# Complex Hessian measures with respect to a background Hermitian form

Sławomir Kołodziej and Ngoc Cuong Nguyen  Faculty of Mathematics and Computer
Science, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland
<EMAIL_ADDRESS>Department of Mathematical Sciences, KAIST,
291 Daehak-ro, Yuseong-gu, Daejeon 34141, South Korea<EMAIL_ADDRESS>
###### Abstract.

We develop potential theory for $m$-subharmonic functions with respect to a
Hermitian metric on a Hermitian manifold. First, we show that the complex
Hessian operator is well-defined for bounded functions in this class. This
allows to define the $m$-capacity and then showing the quasi-continuity of
$m$-subharmonic functions. Thanks to this we derive other results parallel to
those in pluripotential theory such as the equivalence between polar sets and
negligible sets. The theory is then used to study the complex Hessian equation
on compact Hermitian manifold with boundary, with the right hand side of the
equation admitting a bounded subsolution. This is an extension of a recent
result of Collins and Picard dealing with classical solutions.

To the memory of Jean-Pierre Demailly

##  1\. Introduction

The $m$-Hessian operator is defined in terms of elementary symmetric
polynomials of degree $m$ of eigenvalues of the Hessian matrix of the given
function. If the degree is equal to the dimension of the space then one deals
with the most important case of the Monge-Ampère operator. One can also
consider more general symmetric functions of eigenvalues. The nonlinear
equations involving such operators will be called in this article Hessian type
equations. They do appear in geometry in problems involving curvatures, like
prescribed the Gauss curvature equation or the Lagrangian mean curvature
equation. The $m$-Hessian equations in $\mathbb{R}^{n}$ were first solved by
Caffarelli-Nirenberg-Spruck [CNS85] for smooth, non-degenerate data. The study
of weak solutions for measures on the right hand side was initiated by
Trudinger and Wang [TrWa97, TrWa99, TrWa02] (see also [Wa09]).

Here we are interested in the complex setting and weak solutions. For smooth
data the first solutions in complex variables were obtained by Vinacua [V88]
and S.Y. Li [Li04] who followed the method of [CNS85]. Błocki [Bl05] adopted
the methods of pluripotential theory (initiated by Bedford and Taylor [BT76,
BT82] in relation to the complex Monge-Ampère equation) to define the action
of the $m$-Hessian operator on non smooth functions and study weak solutions
of the associated equation.

Let $\Omega\subset\mathbb{C}^{n}$ be an open set and $\omega$ is a positive
Hermitian $(1,1)$-form on $\Omega$. Let $1\leq m\leq n$ be an integer and
consider a function $u\in C^{2}(\Omega,\mathbb{R})$. The complex Hessian
operator with respect to $\omega$ acts on $u$ by

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

The operator is elliptic if we restrict ourselves to functions $u$ whose
eigenvectors $\lambda=(\lambda_{1},...,\lambda_{n})$ of the complex Hessian
matrix $[u_{i\bar{j}}]_{1\leq i,j\leq n}$, with respect to $\omega$, belong to
the Gårding cone

$\Gamma_{m}=\left\\{\lambda\in\mathbb{R}^{n}:S_{1}(\lambda)>0,...,S_{m}(\lambda)>0\right\\},$

where $S_{k}(\lambda)$ is the $k$-th elementary symmetric polynomial on
$\lambda$. Such a function is called $m-\omega$-subharmonic (or $m-\omega$-sh
for short).

For $\omega=dd^{c}|z|^{2}$ the standard Kähler form on $\mathbb{C}^{n}$ Błocki
defined non-smooth $m$-subharmonic functions. He showed that the Hessian
operator acting on a bounded $m$-subharmonic function is a well-defined
positive Radon measure, that the operator is stable under decreasing
sequences, and that the homogeneous Dirichlet problem is solvable. The non-
homogeneous one with the right hand side in $L^{p},\ p>n/m,$ was solved by
Dinew and the first author in [DK14].

On a compact Hermitian manifold $(X,\omega)$ the right $m$-Hessian operator to
consider is

$H_{m}(u)=(dd^{c}u+\omega)^{m}\wedge\omega^{n-m},$

or more generally

$H_{m,\alpha}(u)=(dd^{c}u+\alpha)^{m}\wedge\omega^{n-m},$

where $\alpha$ is another $(1,1)$ form.

For $\omega$ Kähler the counterpart of Calabi-Yau theorem was shown by Dinew
and the first author in [DK17], with a use of earlier $C^{2}$ estimates of
Hou-Ma-Wu [HMW10]. Having this result an analogue of pluripotential theory
yields weak solutions (see [DK14]). We refer the readers to [DiL15], [Lu13a,
Lu13b, Lu15] and [LN15] for results in potential theory for $m-\omega$-sh
functions on a compact Kähler manifold.

Our first goal here is to develop potential theory for $m$-subharmonic
functions (with respect to a Hermitian metric) on a Hermitian manifold. The
results often parallel those of pluripotential theory.

Now we assume that $\omega$ is a general Hermitian metric. The complex
$m$-Hessian equation on compact manifolds was solved independently by
Székelyhidi [Sz18] and Zhang [Zha17]. The authors [KN16] obtained weak
continuous solutions for the right hand side in $L^{p},\ p>n/m.$ This
partially motivates the development of potential theory for $m-\omega$-sh
functions on Hermitian manifolds. Unlike in the Kähler case, we have to deal
with the non-zero torsion terms $dd^{c}\omega$ and $d\omega\wedge
d^{c}\omega$. A direct computation shows that for a smooth $m-\omega$-sh
function $u$, $0\leq p\leq n-m-1$ and $k\geq 1$, the form
$(dd^{c}u)^{k}\wedge\omega^{p}$ may not be positive. Those terms appear when
we perform integration by parts. This makes the proofs of basic potential
estimates in the Hermitian setting substantially more difficult. For example,
we need to fully exploit the properties of the positive cone $\Gamma_{m}$, and
show new inequalities on elementary symmetric polynomials to prove the Chern-
Levine-Nirenberg (CLN) inequality [KN16] and a variant of Cauchy-Schwarz
inequality in this paper. This coupled with the uniform convergence allows us
to define the complex Hessian measure of a continuous $m-\omega$-sh function
$u$ as the weak limit of

(1.1) $H_{m}(u):=\lim_{\delta\to 0}H_{m}(u^{\delta})=\lim_{\delta\to
0}(dd^{c}u^{\delta})^{m}\wedge\omega^{n-m},$

where $\\{u^{\delta}\\}$ is a sequence of smooth $m-\omega$-sh functions
converging uniformly to $u$.

However, if $u$ is a bounded $m-\omega$-sh function up till now we have not
been able to define its complex Hessian measure. Thanks to a basic observation
(Lemma 3.1) that $dd^{c}u$ is a current of order zero we found a very natural
way to extend the classical approach of Bedford-Taylor in this case. The first
main result of the paper (Theorem 3.3) shows that the measure on the left hand
side of (1.1) can be defined by taking weak limit inductively. Thus, for
bounded $m-\omega$-subharmonic functions $u$ and $u^{\delta}\downarrow u$
point-wise as $\delta\to 0$,

$H_{m}(u):=\lim_{\delta_{m}\to 0}\cdots\lim_{\delta_{1}\to
0}dd^{c}u^{\delta_{m}}\wedge\cdots\wedge
dd^{c}u^{\delta_{1}}\wedge\omega^{n-m}$

exists in the sense of currents of order zero. The proof is based on the CLN
inequality in [KN16]. This is the starting point for proving analogues of
Bedford-Taylor results presented in Chapter 1 of [Ko05]. The main difference
is that we no longer have a nice integration by parts formula for closed
positive currents. Instead we need to work with the (non-closed) currents of
order zero in general. This makes our proof of weak convergence of currents
under decreasing limits of $m-\omega$-sh bounded functions in Proposition 3.20
more complicated than the one in [BT82]. Next, we obtain (Theorem 4.9) the
quasi-continuity of $m-\omega$-sh functions with respect to a suitable
$m$-capacity: for a Borel set $E\subset\Omega$,

(1.2) $cap_{m}(E)=\sup\left\\{\int_{E}H_{m}(u):u\text{ is }m-\omega\text{-sh
in }\Omega,-1\leq u\leq 0\right\\}.$

To define this capacity we needed Theorem 3.3. Once quasi-continuity is
proven, one obtains weak convergence of mixed wedge products of the forms
$dd^{c}u_{j}$ for $m-\omega$-sh functions $u_{j}$ under monotone convergence
(Lemma 5.1 and Lemma 5.4) following the classical arguments in [BT82, BT87].
Next we study the polar sets and negligible sets of $m-\omega$-sh functions.
In this setting it seems impossible to obtain nice formulae for the capacity
of compact or open sets in terms of Hessian measures of extremal functions as
it is the case for Monge-Ampère measures. Exploiting further the properties of
$\Gamma_{m}$ in Section 2.4, especially Proposition 2.15 we are able to
compare the outer capacity and the Hessian measures of relative extremal
functions in Lemma 7.5. This suffices to give a characterization of a polar
set by $cap_{m}^{*}(E)=0$ in Proposition 7.7. Consequently, we conclude the
equivalence of polar sets and negligible sets (Theorem 7.8).

In the last sections we apply the above results to the complex $m$-Hessian
equation. Recently, there is a lot of active research on fully non-linear
elliptic equations on compact Hermitian manifolds with or without boundary
(cf. [CM21], [CP22], [D21], [DL21] [Gu98], [GL10], [GLu23], [GP22], [LS20],
[PT21], [Sz18], [STW17], [TW10], [Yu21]) in various geometric contexts.

In particular Collins and Picard [CP22] solved the Dirichlet problem for the
$m$-Hessian equation in an open subset of a Hermitian manifold under the
hypothesis of existence of a subsolution and smooth data. We extend it in
Sections 8, 9 showing that the existence of a bounded subsolution implies the
existence of a bounded solution for both bounded domains (Theorem 8.7) and
compact complex manifolds with boundary (Theorem 9.1). In the proof the above
equivalence of polar and negligible sets will play a crucial role (see Lemma
8.3). The homogeneous $m$-Hessian equation on a (Kähler) manifold with
boundary was recently solved in a particular case in [W23] in relation to the
Wess-Zumino-Witten type equation proposed by Donaldson in [Do99].

Acknowledgement. The first author is partially supported by grant no.
2021/41/B/ST1/01632 from the National Science Center, Poland. The second
author is partially supported by the National Research Foundation of Korea
(NRF) grant no. 2021R1F1A1048185. This project was initiated during the visit
of the first author in KAIST, and he wishes to express his gratitude for great
hospitality and prefect working conditions.

We are very grateful to Zywomir Dinew for kindly pointing out to us the gap in
the proof of Lemma 3.2 in the previous version.

##  2\. Generalized $m$-subharmonic functions

###  2.1. Elementary symmetric positive cones

In this section we prove important point-wise estimates for elementary
symmetric polynomials. Let $1\leq m\leq n$ be two integers. The positive cone
$\Gamma_{m}$ is given by

(2.1)
$\Gamma_{m}=\\{\lambda=(\lambda_{1},...,\lambda_{n})\in\mathbb{R}^{n}:S_{1}(\lambda)>0,...,S_{m}(\lambda)>0\\},$

where $S_{k}(\lambda)=\sum_{1\leq i_{1}<...<i_{k}\leq
n}\lambda_{i_{1}}\cdots\lambda_{i_{k}}$; and conventionally:
$S_{0}(\lambda)=1$ and $S_{k}(\lambda)=0$ for $k<0$ or $k>n$. Let
$\lambda=(\lambda_{1},...,\lambda_{n})\in\Gamma_{m}$ be arranged in the
decreasing manner, i.e.,

$\lambda_{1}\geq\cdots\geq\lambda_{m}\geq\cdots\geq\lambda_{n}.$

Then, we know from [Iv83, Lemma 8] that $\lambda_{m}>0$ which is a consequence
of a characterization of the cone $\Gamma_{m}$. Namely, for
$\\{i_{1},...,i_{t}\\}\subset\\{1,...,n\\}$ such that $k+t\leq m$, one has

(2.2) $S_{k;i_{1}\cdots i_{t}}(\lambda)>0,$

where $S_{k;i_{1}\cdots
i_{t}}(\lambda)=S_{k}|_{\lambda_{i_{1}}=\cdots=\lambda_{i_{t}}=0}.$ This
implies also that for $1\leq k\leq m$,

(2.3) $S_{k-1}(\lambda)\geq\lambda_{1}\cdots\lambda_{k-1}.$

######  Lemma 2.1.

There exists $\theta=\theta(n,m)>0$ such that the following statements hold.

  * (a)

for $1\leq j\leq m$, $\lambda_{j}S_{m-1;j}\geq\theta S_{m};$

  * (b)

for $1\leq i\leq m-1$, $\lambda_{i}S_{m-2;im}\geq\theta S_{m-1;m}.$

###### Proof.

The item (a) follows from [KN16, Eq. (2.7)], while (b) follows from (a) if we
replace $n,m$ and $S_{m}$ by $n-1,m-1$ and $S_{m-1;m}$, respectively. ∎

######  Lemma 2.2.

There exists a uniform constant $C$, depending on $n,m$, such that the
following inequalities are satisfied.

  * (a)

For $1\leq i\leq m-1$ and $\lambda\in\Gamma_{m}$

$\frac{\lambda_{1}\cdots\lambda_{m}}{\lambda_{i}}\leq
C(S_{m-1;i}\,S_{m-1})^{\frac{1}{2}}.$

  * (b)

Generally, for $1\leq\ell\leq n$ and increasing multi-indices
$(i_{1},...,i_{m-1})$,

$\prod_{i_{s}\neq\ell;s=1}^{m-1}|\lambda_{i_{s}}|\leq
C\left(S_{m-1;\ell}S_{m-1}\right)^{\frac{1}{2}}.$

###### Proof.

(a) The inequality is equivalent to saying that there exist uniform constants
$c_{1},c_{2}>0$ such that for every positive number $a$,

(2.4) $a\frac{\lambda_{1}\cdots\lambda_{m}}{\lambda_{i}}\leq
c_{1}a^{2}S_{m-1;i}+c_{2}S_{m-1},$

where $1\leq i\leq m-1$. In fact as we will see in the proof $c_{1},c_{2}$ are
explicitly given constants. We observe that if $a\leq 1$, then we can easily
get the claim as

$\lambda_{1}\cdots\lambda_{m}/\lambda_{i}\leq\lambda_{1}\cdots\lambda_{m-1}\leq
S_{m-1}.$

Now we consider $a>1$. We prove (a) for the case $i=1$, the other cases $1\leq
i\leq m-1$ follow in the same way. The basic identities/inequalities are

(2.5) $\displaystyle S_{m;1m}+\lambda_{1}S_{m-1;1m}=S_{m;m}$
$\displaystyle=S_{m}-\lambda_{m}S_{m-1;m}$
$\displaystyle\geq-\lambda_{m}S_{m-1;m},$

and

(2.6) $S_{m-1;1m}+\lambda_{1}S_{m-2;1m}=S_{m-1;m}.$

Multiplying (2.5) by $S_{m-2;1m}$ and (2.6) by $S_{m-1;1m}$ to eliminate
$\lambda_{1}$ we get that

(2.7) $\displaystyle S_{m-1;1m}^{2}-S_{m;1m}S_{m-2;1m}$ $\displaystyle\leq
S_{m-1;m}(S_{m-1;1m}+\lambda_{m}S_{m-2;1m})$
$\displaystyle=S_{m-1;m}S_{m-1;1}.$

The Newton inequality holds for every $\lambda\in\mathbb{R}^{n}$ and tells us

$S_{m;1m}S_{m-2;1m}\leq\frac{(m-1)(n-m+1)}{m(n-m+2)}[S_{m-1;1m}]^{2}=:c_{m}[S_{m-1;1m}]^{2}.$

Notice that $0<c_{m}<1$. Hence, we derive from the above and (2.7) that

$S_{m-1;1m}^{2}-c_{m}[S_{m-1;1m}]^{2}\leq S_{m-1;m}S_{m-1;1}.$

Therefore,

(2.8) $S_{m-1;1m}^{2}\leq\frac{1}{1-c_{m}}S_{m-1;m}S_{m-1;1}.$

Using Cauchy-Schwarz’ inequality, the inequality (2.8) and the formula

$S_{m-1;1m}=S_{m-1;1}-\lambda_{m}S_{m-2;1m},$

we get

$\displaystyle a^{2}S_{m-1;1}+\frac{1}{4(1-c_{m})}S_{m-1;m}$
$\displaystyle\geq
a\left[\frac{S_{m-1;1}S_{m-1;m}}{1-c_{m}}\right]^{\frac{1}{2}}$
$\displaystyle\geq a|S_{m-1;1m}|$ $\displaystyle\geq
a(-S_{m-1;1}+\lambda_{m}S_{m-2;1m}).$

This implies the inequality

(2.9) $(a^{2}+a)S_{m-1;1}+\frac{1}{4(1-c_{m})}S_{m-1;m}\geq
a\lambda_{m}S_{m-2;1m}.$

Since $a\geq 1$, it follows that $2a^{2}S_{m-1;1}+CS_{m-1}\geq
a\lambda_{m}S_{m-2;1m}.$ So, using Lemma 2.1 and (2.3) we have

(2.10) $\displaystyle(2a^{2}S_{m-1;1}+CS_{m-1})\lambda_{1}$ $\displaystyle\geq
a\lambda_{m}\lambda_{1}S_{m-2;1m}$ $\displaystyle\geq
a\theta^{2}\lambda_{m}S_{m-1}$ $\displaystyle\geq
a\theta^{2}\lambda_{1}\cdots\lambda_{m}.$

The proof of the lemma is completed with $c_{1}=2$ and
$c_{2}=\frac{1}{4(1-c_{m})}$ and $C=\sqrt{2/(1-c_{m})}.$

(b) The characterization (2.2) implies that a sum of any $(n-m+1)$ entries of
$\lambda$ is positive. Hence, we have for $\lambda_{i_{s}}\leq 0$,

$|\lambda_{i_{s}}|\leq(n-m)\lambda_{m}.$

So, for $1\leq\ell\leq m-1$,

(2.11)
$\displaystyle\prod_{i_{s}\neq\ell;s=1}^{m-1}|\lambda_{i_{s}}|\leq(n-m)^{m-1}\frac{\lambda_{1}\cdots\lambda_{m}}{\lambda_{\ell}}\leq
C\left[S_{m-1;\ell}S_{m-1}\right]^{\frac{1}{2}},$

where we used (a) for the second inequality.

Now we treat the remaining range $m\leq\ell\leq n$. By a result of Lin and
Trudinger [LT94, Theorem 1.1], we know that $S_{m-1;\ell}\geq\theta S_{m-1}$
for a constant $\theta=\theta(n,m)$ depending only on $n,m$. This implies

$S_{m-1}\leq(S_{m-1;\ell}S_{m-1})^{\frac{1}{2}}/\sqrt{\theta}.$

Thus, the desired inequality easily follows from this and the bound

(2.12)
$\displaystyle\prod_{i_{s}\neq\ell;s=1}^{m-1}|\lambda_{i_{s}}|\leq(n-m)^{m-1}\lambda_{1}\cdots\lambda_{m-1}$

for $m\leq\ell\leq n$. The proof of (b) is completed. ∎

###  2.2. Cauchy-Schwarz’s inequality

Let $\omega$ be a Hermitian metric on $\mathbb{C}^{n}$ and let $\Omega$ be a
bounded open set in $\mathbb{C}^{n}$. The positive cone $\Gamma_{m}(\omega)$,
associated to $\omega$, of real $(1,1)$-forms is defined as follows. A real
$\gamma$ is said to belong $\Gamma_{m}(\omega)$ if at any point $z\in\Omega$,

$\gamma^{k}\wedge\omega^{n-k}(z)>0\quad\text{for }k=1,...,m.$

Equivalently, in the normal coordinate system with respect to $\omega$ at $z$,
diagonalizing $\gamma=\sqrt{-1}\sum_{i}\lambda_{i}dz_{i}\wedge d\bar{z}_{i}$,
we have $\lambda=(\lambda_{1},...,\lambda_{n})\in\Gamma_{m}$. Now we will
translate the estimates in Section 2.1 into the integral forms.

We can state the following versions of Cauchy-Schwarz’s inequality in this
setting. Let $h$ be a smooth real-valued function and $\phi,\psi$ be Borel
functions. Let $T$ be a positive current of bidegree $(n-2,n-2)$.

######  Lemma 2.3.

There exists a uniform constant $C$ depending on $\omega$ such that

$\displaystyle\left|\int\phi\psi\;dh\wedge d^{c}\omega\wedge T\right|^{2}$
$\displaystyle\leq C\int|\phi|^{2}\;dh\wedge d^{c}h\wedge\omega\wedge
T\int|\psi|^{2}\;\omega^{2}\wedge T.$

###### Proof.

The proof of [Ng16, Proposition 1.4] can be easily adapted. ∎

This lemma can be applied for the case $T=\gamma^{s}\wedge\omega^{n-m+\ell}$,
where $\gamma\in\Gamma_{m}(\omega)$ and $0\leq s,\ell\leq m-1$ and
$s+\ell=m-1$. We also need to deal with possible non-positive forms
$T^{\prime}=\gamma^{m-1}\wedge\omega^{n-m-1}$ where the classical Cauchy-
Schwarz is not immediately applicable. However, we still have

######  Lemma 2.4.

There exists a uniform constant $C$ depending on $\omega,n,m$ such for every
$\gamma\in\Gamma_{m}(\omega)$,

$\displaystyle\left|\int\phi\psi\;dh\wedge
d^{c}\omega\wedge\gamma^{m-1}\wedge\omega^{n-m-1}\right|^{2}$
$\displaystyle\leq C\int|\phi|^{2}\;dh\wedge
d^{c}h\wedge\gamma^{m-1}\wedge\omega^{n-m}\times\int|\psi|^{2}\;\gamma^{m-1}\wedge\omega^{n-m+1}.$

###### Proof.

We express the integrands of both sides as follows.

$\displaystyle dh\wedge
d^{c}\omega\wedge\gamma^{m-1}\wedge\omega^{n-m-1}=f_{1}(z)\omega^{n},$
$\displaystyle dh\wedge
d^{c}h\wedge\gamma^{m-1}\wedge\omega^{n-m}=[f_{2}(z)]^{2}\omega^{n},$
$\displaystyle\gamma^{m-1}\wedge\omega^{n-m+1}=[f_{3}(z)]^{2}\omega^{n}.$

Thus, the inequality will follow from the classical Cauchy-Schwarz inequality
if we have point-wise $|f_{1}(z)|\leq Cf_{2}(z)f_{3}(z)$ for every
$z\in\Omega$. This is proved by using the normal coordinate system at a given
point $z$ with respect to $\omega$ which diagonalizes also $\gamma$, i.e.,

$\omega=\sqrt{-1}\sum_{i=1}^{n}dz_{i}\wedge
d\bar{z}_{i},\quad\gamma=\sqrt{-1}\sum_{i}\lambda_{i}dz_{i}\wedge
d\bar{z}_{i},$

where $\lambda=(\lambda_{1},...,\lambda_{n})\in\Gamma_{m}$. Denote
$h_{i}=\partial h/\partial z_{i}$. Then, at the point $z$,

(2.13)
$\binom{n}{m-1}(f_{2})^{2}=\sum_{i=1}^{n}|h_{i}|^{2}S_{m-1;i},\quad\binom{n}{m-1}(f_{3})^{2}=S_{m-1}.$

Now, observe that $\gamma^{m-1}\wedge\omega^{n-m-1}$ is a $(n-2,n-2)$ form, so
after taking the wedge product with $dh\wedge d^{c}\omega$ the non-zero
contribution give only $\sqrt{-1}\partial h\wedge\overline{\partial}\omega$
and $\sqrt{-1}\;\overline{\partial}h\wedge\partial\omega$. As $h$ is a real
valued function, these two forms are mutually conjugate. Let us write

$\overline{\partial}\omega=\sum\omega_{i\bar{j}\bar{k}}d\bar{z}_{k}\wedge
dz_{i}\wedge d\bar{z}_{j}.$

Denote $dV=(\sqrt{-1})^{n^{2}}dz_{1}\wedge\cdots dz_{n}\wedge
d\bar{z}_{1}\wedge\cdots d\bar{z}_{n}.$ Let $J=(j_{1},...,j_{n-m-1})$ be an
increasing multi-index. Then,

$\frac{1}{(m-1)!}\partial
h\wedge\overline{\partial}\omega\wedge\gamma^{m-1}\wedge dz_{J}\wedge
d\bar{z}_{J}/dV=\sum_{j,\ell\not\in
J;j\neq\ell}c_{j\bar{j}\bar{\ell}}\;h_{\ell}\prod_{i_{s}\not\in
J\cup\\{j,\ell\\}}\lambda_{i_{s}},$

where $c_{i\bar{i}\bar{\ell}}$ is $\omega_{i\bar{i}\bar{\ell}}$ or
$\omega_{i\bar{\ell}\,\bar{i}}$, and $i_{s}\in I=(i_{1},..,i_{m-1})$ which is
an increasing multi-index satisfying $I\cap J=\emptyset$. Then, at the point
$z$,

(2.14) $|f_{1}(z)|\leq
c_{0}\sum_{|I|=m-1}\sum_{\ell=1}^{n}|h_{\ell}|\prod_{i_{s}\neq\ell;s=1}^{m-1}|\lambda_{i_{s}}|,$

where $c_{0}$ is a uniform constant depending only on $\omega$.

By (2.13) and (2.14) we have reduced the inequality $|f_{1}|\leq Cf_{2}f_{3}$
to the one for symmetric polynomials. To show the latter, by Lemma 2.2-(b) for
$1\leq\ell\leq n$, we have

$\displaystyle|h_{\ell}|\prod_{i_{s}\neq\ell;s=1}^{m-1}|\lambda_{i_{s}}|$
$\displaystyle\leq
C\left[|h_{\ell}|^{2}S_{m-1;\ell}S_{m-1}\right]^{\frac{1}{2}}$
$\displaystyle\leq
C\left(\sum_{i=1}^{n}|h_{i}|^{2}S_{m-1;i}\right)^{\frac{1}{2}}[S_{m-1}]^{\frac{1}{2}}.$

Taking the sum of (finitely many) terms on the left hand side of (2.14) the
proof of the theorem follows. ∎

We also need this inequality for wedge products of two forms and more. This is
done by solving a linear system of inequalities as in [KN16, page 2226].

######  Corollary 2.5.

There exists a uniform constant $C$, depending on $\omega,n,m$, such that the
following inequalities hold.

  * (a)

For $\eta,\gamma\in\Gamma_{m}(\omega)$,

$\displaystyle\left|\int\phi\psi dh\wedge
d^{c}\omega\wedge\eta^{k}\wedge\gamma^{m-k-1}\wedge\omega^{n-m-1}\right|^{2}$
$\displaystyle\leq C\int|\phi|^{2}dh\wedge
d^{c}h\wedge(\eta+\gamma)^{m-1}\wedge\omega^{n-m}\int|\psi|^{2}(\eta+\gamma)^{m-1}\wedge\omega^{n-m+1}.$

  * (b)

Generally, for $\gamma_{1},...,\gamma_{m-1}\in\Gamma_{m}(\omega)$,

$\displaystyle\left|\int\phi\psi dh\wedge
d^{c}\omega\wedge\gamma_{1}\wedge\cdots\wedge\gamma_{m-1}\wedge\omega^{n-m-1}\right|^{2}$
$\displaystyle\leq C\int|\phi|^{2}\;dh\wedge
d^{c}h\wedge(\sum_{i=1}^{m-1}\gamma_{i})^{m-1}\wedge\omega^{n-m}\times$
$\displaystyle\qquad\times\int|\psi|^{2}(\sum_{i=1}^{m-1}\gamma_{i})^{m-1}\wedge\omega^{n-m+1}.$

###  2.3. $m$-subharmonic functions on Hermitian manifolds

Let us recall the definition of generalized $m$-subharmonic function in the
Hermitian setting (see [Bl05], [KN16] and [GN18]). Let $\Omega$ be a bounded
open set in $\mathbb{C}^{n}$ and let $\omega$ be a Hermitian metric on
$\Omega$.

######  Definition 2.6.

An upper semi-continuous function $u:\Omega\to[-\infty,+\infty)$ is called
$m-\omega$-subharmonic if $u\in L^{1}_{\rm loc}(\Omega)$ and for any
collection $\gamma_{1},...,\gamma_{m-1}\in\Gamma_{m}(\omega)$,

$dd^{c}u\wedge\gamma_{1}\wedge\cdots\wedge\gamma_{m-1}\wedge\omega^{n-m}\geq
0$

in the sense of currents.

######  Remark 2.7.

By Gårding’s [Ga59] results a function $u\in C^{2}(\Omega)$ is $m-\omega$-sh
if and only if $dd^{c}u\in\overline{\Gamma_{m}(\omega)}$, that is
$dd^{c}u\wedge\gamma_{1}\wedge\cdots\wedge\gamma_{m-1}\geq 0$ point-wise in
$\Omega$. Thus, the estimates for forms in $\Gamma_{m}(\omega)$ are applicable
to $dd^{c}u$ if $u$ is a smooth $m-\omega$-sh function.

It follows from Michelsohn [Mi82] that for
$\gamma_{1},...,\gamma_{m-1}\in\Gamma_{m}(\omega)$ there is a unique $(1,1)$
positive form $\alpha$ such that

$\alpha^{n-1}=\gamma_{1}\wedge\cdots\wedge\gamma_{m-1}\wedge\omega^{n-m}.$

The above definition of $m-\omega$-sh function can be expressed more
familiarly, in terms of potential theory, using the notion of
$\alpha$-subharmonicity (see e.g., [GN18, Definition 2.1, Lemma 9.10]). Thanks
to this many potential-theoretic properties of $m-\omega$-sh functions can be
derived from those of $\alpha$-sh functions. For example, if two $m-\omega$-sh
functions are equal almost everywhere (with respect to the Lebesgue measure),
then they are equal everywhere [GN18, Corollary 9.7]. One also has the ”gluing
property” that allows to modify the function outside a compact subset. This is
an immediate consequence of [GN18, Lemma 9.5].

######  Lemma 2.8.

Let $U\subset\Omega$ be two open sets in $\mathbb{C}^{n}$. Let $u$ be
$m-\omega$-sh in $U$ and $v$ be $m-\omega$-sh in $\Omega$. Assume that
$\limsup_{z\to x}u(z)\leq v(x)$ for every $x\in\partial U\cap\Omega$. Then,
the function

$\widetilde{u}=\begin{cases}\max\\{u,v\\}&\quad\text{on }U,\\\ v&\quad\text{on
}\Omega\setminus U,\end{cases}$

is $m-\omega$-sh in $\Omega$.

Because of this we have the following way of reducing a proof to a simpler
case (see [Ko05, page 7] for the proof).

######  Localization Principle.

If we are to prove the weak convergence or local estimate for a family of
locally uniformly bounded $m-\omega$-sh functions, it is no loss of generality
to we assume that the functions are defined in a ball and are all equal on
some neighborhood of the boundary.

For a bounded psh function its convolution with a radial smoothing kernel
provides locally a smooth, decreasing approximation of this function. It is no
longer the case for generalized $m-\omega$-sh. However, in a strictly
$m$-pseudoconvex domain $\Omega$, that is for $\Omega=\\{\rho<0\\}$, where
$\rho\in C^{2}(\overline{\Omega})$ is strictly $m-\omega$-sh and $d\rho\neq 0$
on $\partial\Omega$, we still have (non-explicit) way of approximation by
smooth $m-\omega$-sh functions.

######  Proposition 2.9.

Let $\Omega\subset\subset\mathbb{C}^{n}$ be strictly $m$-pseudoconvex domain.
Let $u$ be a bounded $m-\omega$-sh function in a neighborhood of
$\overline{\Omega}$. Then, there exists a sequence of smooth $m-\omega$-sh
functions $u_{j}\in C^{\infty}(\overline{\Omega})$ that decreases to $u$
point-wise in $\overline{\Omega}$.

###### Proof.

The proof is the same as the one of [GN18, Theorem 3.18] if we replace the
ball by a strictly $m$-pseudoconvex domain and invoke [CP22, Theorem 1.1] for
the smooth solution of the Dirichlet problem on a strictly $m$-pseudoconvex
domain. ∎

###  2.4. Integral estimates for smooth functions

Let $\Omega$ be a bounded open set in $\mathbb{C}^{n}$. Let $-1\leq v\leq
w\leq 0$ be smooth $m-\omega$-sh functions in $\Omega$ such that $v=w$ in a
neighborhood of $\partial\Omega$. Let $\rho$ be a smooth $m-\omega$-sh
function such that $-1\leq\rho\leq 0$. Using the notation
$\gamma:=dd^{c}\rho$, $h=w-v$ we consider

(2.15) $e_{(q,k,s)}=\int
h^{q+1}\gamma^{k}\wedge(dd^{c}v)^{s}\wedge\omega^{n-k-s},$

where $q\geq 0$, the integers $0\leq k\leq m$ and $0\leq s\leq m-k$. We wish
to bound

$e_{(q,m,0)}=\int h^{q+1}\gamma^{m}\wedge\omega^{n-m},$

by the integrals

$e_{(r,0,i)}=\int h^{r+1}(dd^{c}v)^{i}\wedge\omega^{n-i},$

where $i=0,...,m$ and $1\leq r<q$. This is done via repeated use of the
integration by parts to replace $\gamma$ by $dd^{c}v$. However, there are
three different cases depending on the total degree $k+s$ of the form
$\gamma^{k}\wedge(dd^{c}v)^{s}$ that we need to deal with separately.

  * •

Case 1: $k+s=m$;

  * •

Case 2: $k+s=m-1$;

  * •

Case 3: $k+s\leq m-2$.

Let us start with an auxiliary inequality that we use frequently bellow.

######  Lemma 2.10.

Let $p\geq 1$ and $0\leq k\leq m-1$. There exists a constant $C$ depending on
$\omega,n,m$ such that

  * (a)

for $0\leq s\leq m-1-k$:

$\displaystyle\int h^{p-1}dh\wedge
d^{c}h\wedge\gamma^{k}\wedge(dd^{c}v)^{s}\wedge\omega^{n-k-s-1}$
$\displaystyle\leq\int
h^{p}\gamma^{k}\wedge(dd^{c}v)^{s+1}\wedge\omega^{n-k-s-1}+C\int
h^{p+1}(\gamma+dd^{c}v)^{k+s}\wedge\omega^{n-k-s}.$

  * (b)

for $0\leq s\leq m-3-k$:

$\displaystyle\int h^{p-1}dh\wedge
d^{c}h\wedge\gamma^{k}\wedge(dd^{c}v)^{s}\wedge\omega^{n-k-s-1}$
$\displaystyle\leq\int
h^{p}\gamma^{k}\wedge(dd^{c}v)^{s+1}\wedge\omega^{n-k-s-1}+C\int
h^{p+1}\gamma^{k}\wedge(dd^{c}v)^{s}\wedge\omega^{n-k-s}.$

###### Proof.

(a) Note first that $0\leq h\leq 1$ and also
$T:=\gamma^{k}\wedge(dd^{c}v)^{s}\wedge\omega^{n-k-s-1}$ and $dd^{c}w\wedge T$
are positive forms for $n-s-k-1\geq n-m$. Therefore,

$\displaystyle p(p+1)h^{p-1}dh\wedge d^{c}h\wedge T$
$\displaystyle=[dd^{c}h^{p+1}-(p+1)h^{p}dd^{c}h]\wedge T$
$\displaystyle\leq[dd^{c}h^{p+1}+(p+1)h^{p}dd^{c}v]\wedge T.$

Hence,

(2.16) $\displaystyle\int h^{p-1}dh\wedge
d^{c}h\wedge\gamma^{k}\wedge(dd^{c}v)^{s}\wedge\omega^{n-s-k-1}$
$\displaystyle\leq\int(dd^{c}h^{p+1}+h^{p}dd^{c}v)\wedge\gamma^{k}\wedge(dd^{c}v)^{s}\wedge\omega^{n-s-k-1}.$

It remains to estimate the product involving the first term in the bracket. By
integration by parts and [Lemma 2.3, KN16],

(2.17) $\displaystyle\int
dd^{c}h^{p+1}\wedge\omega^{n-s-k-1}\wedge\gamma^{k}\wedge(dd^{c}v)^{s}$
$\displaystyle=\int
h^{p+1}dd^{c}(\omega^{n-s-k-1})\wedge\gamma^{k}\wedge(dd^{c}v)^{s}$
$\displaystyle\leq C\int h^{p+1}(\gamma+dd^{c}v)^{k+s}\wedge\omega^{n-m+1}.$

Combining the last two inequalities the proof of the lemma follows.

(b) The proof is very similar, we first have (2.16). Then, in the middle
integral of (2.17) one can express
$dd^{c}(\omega^{n-s-k-1})=\eta\wedge\omega^{n-m}$ for a smooth
$(m-s-k,m-s-k)$-form $\eta$. Then, since
$\gamma,dd^{c}v\in\Gamma_{m}(\omega)$, in this case the inequality has a
better form

$\left|\int
h^{p+1}\eta\wedge\gamma^{k}\wedge(dd^{c}v)^{s}\wedge\omega^{n-m}\right|\leq
C\int h^{p+1}\gamma^{k}\wedge(dd^{c}v)^{s}\wedge\omega^{n-k-s}.$

The item (b) is proven. ∎

Let us start with the simplest situation in Case 1. We are going to show that

(2.18) $e_{(q,m,0)}\leq C\left(e_{(q-1,m-1,1)}+e_{(q-1,m-1,0)}\right),$

where $C$ is a uniform constant depending on $\omega,m,n$. Equivalently,

######  Lemma 2.11.

Let $q\geq 1$. Then,

$\displaystyle\int_{\Omega}(w-v)^{q+1}\gamma^{m}\wedge\omega^{n-m}$
$\displaystyle\leq C\int_{\Omega}(w-v)^{q}\gamma^{m-1}\wedge
dd^{c}v\wedge\omega^{n-m}$
$\displaystyle\quad+C\int_{\Omega}(w-v)^{q}\gamma^{m-1}\wedge\omega^{n-m+1}.$

###### Proof.

Recall that $h:=w-v\geq 0$ and $h=0$ near $\partial\Omega$. A direct
computation gives

(2.19) $\displaystyle dd^{c}(h^{q+1}\omega^{n-m})$
$\displaystyle=q(q+1)h^{q-1}dh\wedge d^{c}h\wedge\omega^{n-m}$
$\displaystyle\quad+(q+1)h^{q}dd^{c}h\wedge\omega^{n-m}$
$\displaystyle\quad+(q+1)(n-m)h^{q}d\omega\wedge d^{c}h\wedge\omega^{n-m-1}$
$\displaystyle\quad+(q+1)(n-m)h^{q}dh\wedge d^{c}\omega\wedge\omega^{n-m-1}$
$\displaystyle\quad+h^{q+1}dd^{c}\omega^{n-m}$
$\displaystyle\quad=:T_{0}+T_{1}+T_{2}+T_{3}+T_{4}.$

By integration by parts,

(2.20) $\displaystyle\int
h^{q+1}dd^{c}\rho\wedge\gamma^{m-1}\wedge\omega^{n-m}$ $\displaystyle=\int\rho
dd^{c}(h^{q+1}\omega^{n-m})\wedge\gamma^{m-1}$
$\displaystyle=\int\rho(T_{0}+T_{1}+T_{2}+T_{3}+T_{4})\wedge\gamma^{m-1}.$

Case 1a: Estimate of $T_{0}$. Since $-1\leq\rho\leq 0$ and $T_{0}$ is a
positive current,

(2.21) $\rho T_{0}\wedge\gamma^{m-1}\leq 0$

Case 1b: Estimate of $T_{1}$. Similarly, because $v$ is a $m-\omega$-sh
function,

(2.22) $\displaystyle\rho T_{1}\wedge\gamma^{m-1}$ $\displaystyle=(q+1)\rho
h^{q}(dd^{c}w-dd^{c}v)\wedge\omega^{n-m}\wedge\gamma^{m-1}$
$\displaystyle\leq(q+1)h^{q}dd^{c}v\wedge\gamma^{m-1}\wedge\omega^{n-m}.$

Case 1c: Estimate of $T_{4}$. Using the inequality [Lemma 2.3, KN16]

(2.23) $\gamma^{m-1}\wedge dd^{c}\omega^{n-m}\leq
C_{m,n}\gamma^{m-1}\wedge\omega^{n-m+1},$

we can estimate the last term $T_{4}$,

(2.24) $\displaystyle\left|\int\rho h^{q+1}\gamma^{m-1}\wedge
dd^{c}\omega^{n-m}\right|$ $\displaystyle\leq
C\int|\rho|h^{q+1}\gamma^{m-1}\wedge\omega^{n-m+1}$ $\displaystyle\leq
Ce_{(q,m-1,0)},$

where we used again the fact that $|\rho|\leq 1$.

Case 1d: Estimate of $T_{2}$ and $T_{3}$. We use Cauchy-Schwarz’ inequality in
Lemma 2.4, where an extra uniform constant appears. Let us estimate $T_{2}$
(for $T_{3}$ the estimate is completely the same). By Cauchy-Schwarz’
inequality

(2.25) $\displaystyle\left|\int\rho h^{q}d\omega\wedge
d^{c}h\wedge\omega^{n-m-1}\wedge\gamma^{m-1}\right|^{2}$ $\displaystyle\leq
C\int|\rho|h^{q-1}dh\wedge
d^{c}h\wedge\omega^{n-m}\wedge\gamma^{m-1}\int|\rho|h^{q+1}\omega^{n-m+1}\wedge\gamma^{m-1}$
$\displaystyle\leq C\left(\int h^{q-1}dh\wedge
d^{c}h\wedge\gamma^{m-1}\wedge\omega^{n-m}+\int
h^{q+1}\gamma^{m-1}\wedge\omega^{n-m+1}\right)^{2}$ $\displaystyle\leq
C\left(e_{(q-1,m-1,1)}+e_{(q,m-1,0)}\right)^{2}$

where we used Lemma 2.10 with $s=0,k=m-1$ and $p=q$ for the first integral in
the last inequality, namely,

$\int h^{q-1}dh\wedge d^{c}h\wedge\omega^{n-m}\wedge\gamma^{m-1}\leq
C(e_{(q-1,m-1,1)}+e_{(q,m-1,0)}).$

Combining the estimates (2.21), (2.22), (2.24) and (2.25) and the fact that
$e_{(q,\bullet,\bullet)}\leq e_{(q-1,\bullet,\bullet)}\leq
e_{(q-2,\bullet,\bullet)}$ we conclude the proof of the lemma. ∎

We can now state the general inequality in Case 1.

######  Corollary 2.12.

For $1\leq k\leq m$ and $s=m-k\geq 0$ and $q\geq 2$,

$e_{(q,k,s)}\leq
c_{k}\sum_{i=0}^{k-1}e_{(q-1,i,m-i)}+C\sum_{i=0}^{m-1}e_{(q-2,i,m-1-i)}.$

###### Proof.

The proof is by induction in $k$ but ”downward”. For $k=m$ it is Lemma 2.11.
Assume that it is true for every $k+1\leq\ell\leq m$, i.e., we have

(2.26) $e_{(q,\ell,m-\ell)}\leq
c_{\ell}\sum_{i=0}^{k}e_{(q-1,i,m-i)}+c_{\ell}\sum_{i=0}^{m-1}e_{(q-2,i,m-1-i)}.$

We proceed to prove the conclusion holds for $k$. The strategy is the same as
in the proof of the last lemma, however we need to estimate $T_{2}$ and
$T_{3}$ more carefully. We repeat the steps of the proof of Lemma 2.11
replacing $dd^{c}\rho\wedge\Gamma$ by $dd^{c}\rho\wedge\Gamma^{(s)}$, where

$\Gamma=\gamma^{m-1}\wedge\omega^{n-m}\text{ and
}\Gamma^{(s)}=\gamma^{m-1-s}\wedge(dd^{c}v)^{s}\wedge\omega^{n-m},$

in the integrand on the left hand side. The corresponding estimates for
$T_{0},T_{1}$ are similar. Namely,

(2.21′) $\rho T_{0}\wedge\gamma^{m-1-s}\wedge(dd^{c}v)^{s}\leq 0,$

and

(2.22′) $\rho
T_{1}\wedge\gamma^{m-s-1}\leq(q+1)h^{q}(dd^{c}v)^{s+1}\wedge\gamma^{m-1-s}\wedge\omega^{n-m}.$

The one for $T_{4}$ is

(2.23′) $\displaystyle\gamma^{m-1-s}\wedge(dd^{c}v)^{s}\wedge
dd^{c}\omega^{n-m}$ $\displaystyle\leq
C_{m,n}(\gamma+dd^{c}v)^{m-1}\wedge\omega^{n-m+1}$ $\displaystyle\leq
C_{m,n}[\gamma^{m-1}+\gamma^{m-2}\wedge
dd^{c}v+\cdots+(dd^{c}v)^{m-1}]\wedge\omega^{n-m+1}.$

Hence, integrating both sides and using $0\leq h\leq 1$, leads to

(2.24′) $\left|\int h^{q+1}\gamma^{m-1-s}\wedge(dd^{c}v)^{s}\wedge
dd^{c}\omega^{n-m}\right|\leq C\sum_{i=0}^{m-1}e_{(q,i,m-1-i)}.$

Lastly for $T_{2}$ and $T_{3}$, we need to use Corollary 2.5,

$\displaystyle I^{2}:=\left|\int\rho h^{q}d\omega\wedge
d^{c}h\wedge\omega^{n-m-1}\wedge\gamma^{m-1-s}\wedge(dd^{c}v)^{s}\right|^{2}$
$\displaystyle\leq C\int h^{q-1}(\gamma+dd^{c}v)^{m-1}\wedge\omega^{n-m+1}$
$\displaystyle\qquad\times\int h^{q+1}dh\wedge
d^{c}h\wedge(\gamma+dd^{c}v)^{m-1}\wedge\omega^{n-m}$

The standard Cauchy-Schwarz inequality (Lemma 2.3) gives for $\varepsilon>0$
to be determined later,

$\displaystyle I$ $\displaystyle\leq\frac{C}{\varepsilon}\int
h^{q-1}(\gamma+dd^{c}v)^{m-1}\wedge\omega^{n-m+1}$
$\displaystyle\qquad+\varepsilon\int h^{q+1}dh\wedge
d^{c}h\wedge(\gamma+dd^{c}v)^{m-1}\wedge\omega^{n-m}.$

By the last inequality in (2.23′) the first integral is bounded by

$\frac{C}{\varepsilon}\sum_{i=0}^{m-1}e_{(q-2,i,m-1-i)}.$

To bound the second integral we use

$(\gamma+dd^{c}v)^{m-1}\wedge\omega^{n-m}\leq
C_{m,n}\sum_{i=0}^{m-1}\gamma^{i}\wedge(dd^{c}v)^{m-1-i}\wedge\omega^{n-m},$

and then Lemma 2.10-(a) for $k=i,s=m-1-i$ and $p=q+2$. This gives a bound for
the second integral by

$C\varepsilon\sum_{i=0}^{m-1}e_{(q+1,i,m-i)}+C\varepsilon\sum_{i=0}^{m-1}e_{(q+2,i,m-1-i)}.$

Let us consider the first sum above:

$\varepsilon\sum_{i=0}^{m-1}e_{(q+1,i,m-i)}=\varepsilon\sum_{i\geq
k+1}e_{(q+1,i,m-i)}+\varepsilon\sum_{i=0}^{k-1}e_{(q+1,i,m-i)}+\varepsilon
e_{(q+1,k,m-k)}.$

Applying the induction hypothesis (2.26) to the first term on the right, we
derive

$\displaystyle\varepsilon\sum_{i=0}^{m-1}e_{(q+1,i,m-i)}$
$\displaystyle\leq\varepsilon
b_{k}\left(e_{(q,k,m-k)}+\sum_{i=0}^{k-1}e_{(q,i,m-i)}\right)+\varepsilon
b_{k}\sum_{i=0}^{m-1}e_{(q-1,i,m-1-i)}$
$\displaystyle\qquad+\varepsilon\sum_{i=0}^{k-1}e_{(q+1,i,m-i)}+\varepsilon
e_{(q+1,k,m-k)},$

where $b_{k}=c_{m}+\cdots+c_{k+1}$.

Since $e_{(q+2,\bullet,\bullet)}\leq e_{(q+1,\bullet,\bullet)}\leq
e_{(q,\bullet,\bullet)}\leq e_{(q-1,\bullet,\bullet)}$, it follows from the
above estimates that

(2.25′) $\displaystyle I$
$\displaystyle\leq\varepsilon(1+b_{k})\;e_{(q,k,m-k)}+\varepsilon\sum_{i=0}^{k-1}e_{(q-1,i,m-i)}$
$\displaystyle\qquad+[\varepsilon(b_{k}+1)+C/\varepsilon+C\varepsilon]\sum_{i=0}^{m-1}e_{(q-2,i,m-1-i)}.$

Thus, combining (2.21′), (2.22′), (2.24′) and (2.25′) we have

$\displaystyle e_{(q,k,s)}$ $\displaystyle\leq
Ce_{(q-1,k-1,s+1)}+C\sum_{i=0}^{m-1}e_{(q,i,m-1-i)}$
$\displaystyle\quad+\varepsilon(1+b_{k})e_{(q,k,s)}+\varepsilon\sum_{i=0}^{k-1}e_{(q-1,i,m-i)}$
$\displaystyle\quad+[\varepsilon(b_{k}+1)+C/\varepsilon+C\varepsilon]\sum_{i=0}^{m-1}e_{(q-2,i,m-1-i)}.$

Now we can choose $\varepsilon$ so that $\varepsilon(1+b_{k})=1/2$. Since
$s=m-k$, regrouping terms on the right hand side (decreasing the first
parameter in $e_{(q,i,m-1-i)}$ if necessary) we get for possibly larger $C>0$
that

$e_{(q,k,m-k)}/2\leq(C+\varepsilon)\sum_{i=0}^{k-1}e_{(q-1,i,m-i)}+(C+C/\varepsilon)\sum_{i=0}^{m-1}e_{(q-2,i,m-1-i)}.$

The induction step is proven and the lemma follows. ∎

Next, we consider Case 2.

######  Lemma 2.13.

For $1\leq k\leq m-1$ and $s=m-1-k\geq 0$ and $q\geq 1$, we have

$e_{(q,k,s)}\leq
C\left(e_{(q-1,k-1,s+1)}+\sum_{i=0}^{m-2}e_{(q-1,i,m-2-i)}\right).$

###### Proof.

The basic computation using integration by parts that corresponds to (2.19)
starts with

(2.19′′) $dd^{c}(h^{q+1}\omega^{n-m+1}):=T_{0}+T_{1}+T_{2}+T_{3}+T_{4},$

where each term has higher exponent of $\omega$. The estimates for
$T_{0},T_{1}$ are the same as the ones in (2.21′) and (2.22′). Precisely,

(2.21′′) $\rho T_{0}\wedge\gamma^{m-2-s}\wedge(dd^{c}v)^{s}\leq 0,$

and

(2.22′′) $\rho
T_{1}\wedge\gamma^{m-2-s}\leq(q+1)h^{q}\gamma^{m-2-s}\wedge(dd^{c}v)^{s+1}\wedge\omega^{n-m+1}.$

The one for $T_{4}$ is

(2.23′′) $\displaystyle\gamma^{m-2-s}\wedge(dd^{c}v)^{s}\wedge
dd^{c}(\omega^{n-m+1})$ $\displaystyle\leq
C_{m,n}(\gamma+dd^{c}v)^{m-2}\wedge\omega^{n-m+2}$ $\displaystyle\leq
C_{m,n}[\gamma^{m-2}+\gamma^{m-3}\wedge
dd^{c}v+\cdots+(dd^{c}v)^{m-2}]\wedge\omega^{n-m+2}.$

Integrating both sides and using the fact that $0\leq h\leq 1$ yield

(2.24′′) $\left|\int\rho h^{q+1}\gamma^{m-2-s}\wedge(dd^{c}v)^{s}\wedge
dd^{c}(\omega^{n-m+1})\right|\leq C\sum_{i=0}^{m-2}e_{(q,i,m-2-i)}.$

Next, the corresponding inequalities for $T_{2}$ and $T_{3}$ are easier. This
is due to the fact that

$T_{2}\wedge\gamma^{m-2-s}\wedge(dd^{c}v)^{s}=C_{0}h^{q}d\omega\wedge
d^{c}h\wedge\omega^{n-m}\wedge\gamma^{m-2-s}\wedge(dd^{c}v)^{s}.$

Therefore, the classical Cauchy-Schwarz inequality (Lemma 2.3) is sufficient.
Namely,

$\displaystyle I^{2}:=\left|\int\rho h^{q}d\omega\wedge
d^{c}h\wedge\omega^{n-m}\wedge\gamma^{m-2-s}\wedge(dd^{c}v)^{s}\right|^{2}$
$\displaystyle\leq C\int
h^{q}\gamma^{m-2-s}\wedge(dd^{c}v)^{s}\wedge\omega^{n-m+2}$
$\displaystyle\qquad\times\int h^{q}dh\wedge
d^{c}h\wedge\gamma^{m-2-s}\wedge(dd^{c}v)^{s}\wedge\omega^{n-m+1}.$

The Cauchy-Schwarz inequality gives

$\displaystyle I$ $\displaystyle\leq C\int
h^{q}\gamma^{m-2-s}\wedge(dd^{c}v)^{s}\wedge\omega^{n-m+2}$
$\displaystyle\qquad+C\int h^{q}dh\wedge
d^{c}h\wedge\gamma^{m-2-s}\wedge(dd^{c}v)^{s}\wedge\omega^{n-m+1}.$

Here, the first integral in the sum is $e_{(q-1,k-1,s)}$. By applying Lemma
2.10-(a) for $k-1,s$ and $p=q+1$ one gets a bound for the second integral by

$\displaystyle\int
h^{q+1}\gamma^{m-2-s}\wedge(dd^{c}v)^{s+1}\wedge\omega^{n-m+1}+C\int
h^{q+2}(\gamma+dd^{c}v)^{m-2}\wedge\omega^{n-m+2}$ $\displaystyle\leq
e_{(q,k-1,s+1)}+C\sum_{i=0}^{m-2}e_{(q+1,i,m-2-i)}.$

Combining this with the decreasing property in the first parameter of
$e_{(q,\bullet,\bullet)}$ we get

(2.25′′) $\displaystyle I$ $\displaystyle\leq
C[e_{(q-1,k-1,s)}+e_{(q,k-1,s+1)}+\sum_{i=0}^{m-2}e_{(q+1,i,m-2-i)}]$
$\displaystyle\leq C[e_{(q-1,k-1,s+1)}+\sum_{i=0}^{m-2}e_{(q-1,i,m-2-i)}].$

Finally, combining (2.21′′), (2.22′′), (2.24′′) and (2.25′′) one completes the
proof of the lemma. ∎

Lastly, we consider Case 3.

######  Lemma 2.14.

For $1\leq k\leq m-2$ and $0\leq s\leq m-2-k$ and $q\geq 1$ we have

$e_{(q,k,s)}\leq C\left[e_{(q-1,k-1,s+1)}+e_{(q,k-1,s)}\right].$

###### Proof.

We need to estimate
$dd^{c}\rho\wedge\gamma^{k-1}\wedge(dd^{c}v)^{s}\wedge\omega^{n-k-s}$, where
$n-k-s\geq n-m+2$. Then, there is a significant change in basic computation of

(2.19′′′) $dd^{c}(h^{q+1}\omega^{n-k-s})=T_{0}+T_{1}+T_{2}+T_{3}+T_{4},$

where all forms $T_{i}$ contain powers of $\omega$ with the exponent at least
$n-m$. The estimates for $T_{0},T_{1}$ are the same as in Corollary 2.12 and
improved estimates for $T_{2},T_{3}$ are as in Lemma 2.13. Moreover the bound
for $T_{4}$ is easier. Namely, since
$\gamma^{k-1}\wedge(dd^{c}v)^{s}\wedge\omega^{n-m}$ is a positive form, one
obtains

(2.23′′′) $\gamma^{k-1}\wedge(dd^{c}v)^{s}\wedge dd^{c}(\omega^{n-k-s})\leq
C\gamma^{k-1}\wedge(dd^{c}v)^{s}\wedge\omega^{n-k-s+1}.$

Hence, multiplying both sides by $\rho h^{q+1}$ and then integrating we get

(2.24′′′) $\left|\int\rho h^{q+1}\gamma^{k-1}\wedge(dd^{c}v)^{s}\wedge
dd^{c}(\omega^{n-k-s})\right|\leq Ce_{(q,k-1,s)},$

where we used the fact that $-1\leq\rho\leq 0$.

Next, the estimates for $T_{2}$ and $T_{3}$ are

$\displaystyle I$ $\displaystyle:=\left|\int\rho h^{q}d\omega\wedge
d^{c}h\wedge\omega^{n-k-s-1}\wedge\gamma^{k-1}\wedge(dd^{c}v)^{s}\right|$
$\displaystyle\leq C\int
h^{q+1}\gamma^{k-1}\wedge(dd^{c}v)^{s}\wedge\omega^{n-k-s}$
$\displaystyle\qquad+C\int h^{q-1}dh\wedge
d^{c}h\wedge\gamma^{k-1}\wedge(dd^{c}v)^{s}\wedge\omega^{n-k-s}.$

Using Lemma 2.10-(b) for $k-1,s$ and $p=q-1$ in the last integral above yields

(2.25′′′) $\displaystyle I$ $\displaystyle\leq
Ce_{(q,k-1,s)}+C[e_{(q-1,k-1,s+1)}+e_{(q,k-1,s)}]$ $\displaystyle\leq
C[e_{(q-1,k-1,s+1)}+e_{(q,k-1,s)}].$

Combining the estimates for $T_{0},T_{1}$ in (2.19′′′), (2.24′′′) and
(2.25′′′) we complete the proof of lemma. ∎

We are ready to state the main inequality.

######  Proposition 2.15.

Let $e_{(q,k,s)}$ be the numbers defined by (2.15). Then, for $q=3m$,

$e_{(q,m,0)}\leq C\sum_{s=0}^{m}e_{(0,0,s)},$

where $C=C(\omega,n,m)$ is a uniform constant.

###### Proof.

We start with Lemma 2.11 which gives

(2.27) $e_{(q,m,0)}\leq C\left[e_{(q-1,m-1,1)}+e_{(q-1,m-1,0)}\right].$

Then, the first term in the bracket is estimated via Corollary 2.12. Applying
this corollary $(m-1)$-times and using decreasing property of $e_{(p,k,s)}$ in
the first parameter, we get

(2.28) $e_{(q-1,m-1,1)}\leq
Ce_{(q-m,0,m)}+C\sum_{i=0}^{m-1}e_{(q-m-2,i,m-1-i)}.$

The second term in the bracket in (2.27) satisfies $e_{(q-1,m-1,0)}\leq
e_{(q-m-2,m-1,0)}$. Next, we use Lemma 2.13 for each term
$e_{(q^{\prime},\ell,m-1-\ell)}$ with $q^{\prime}=q-m-2$ in the sum above.
Namely, applying the lemma $\ell$-times and using the decreasing property
again, we get

$e_{(q^{\prime},\ell,m-1-\ell)}\leq
Ce_{(q^{\prime}-\ell,0,m-1)}+C\sum_{i=0}^{m-2}e_{(q^{\prime}-\ell-1,i,m-2-i)}.$

Note that the smallest value of the first parameter in the last sum is
$q^{\prime}-m$ for $\ell=m-1$. Hence,

(2.29) $\sum_{i=0}^{m-1}e_{(q^{\prime},i,m-1-i)}\leq
Ce_{(q^{\prime}-\ell,0,m-1)}+C\sum_{i=0}^{m-2}e_{(q^{\prime}-m,i,m-2-i)}.$

It remains to apply Lemma 2.14 for each term
$e_{(q^{\prime\prime},\ell,m-2-\ell)}$ in the sum on the right hand side,
where $q^{\prime\prime}=q^{\prime}-m=q-2m-2$. Again, we have

$\displaystyle e_{(q^{\prime\prime},\ell,m-2-\ell)}$ $\displaystyle\leq
Ce_{(q^{\prime\prime}-1,\ell-1,m-\ell-1)}+Ce_{(q^{\prime\prime},\ell-1,m-\ell-2)}$
$\displaystyle\leq\cdots$ $\displaystyle\leq
Ce_{(q^{\prime\prime}-\ell,0,m-2)}+\sum_{i=0}^{\ell-1}e_{(q^{\prime\prime}-i,\ell-1-i,m-\ell-2+i)}.$

Therefore, an easy induction argument gives us

(2.30) $e_{(q^{\prime\prime},\ell,m-2-\ell)}\leq
C\sum_{s=0}^{m-2}e_{(q^{\prime\prime}-\ell,0,s)}.$

Combining (2.27), (2.28), (2.29) and (2.30) we arrive at

$e_{(q,m,0)}\leq
C\sum_{s=0}^{m}e_{(q^{\prime\prime}-m+2,0,s)}=C\sum_{s=0}^{m}e_{(0,0,s)}$

as $q^{\prime\prime}-m+2=q-3m=0.$ ∎

So far all considered functions were smooth, however, by [KN16, Proposition
2.11] we know that the integrands on both sides of the above statements
(Lemmas 2.10-2.14, Corollary 2.12 and Proposition 2.15) are well-defined for
continuous $m-\omega$-sh functions. Let us record the following observation.

######  Remark 2.16.

Let $\Omega$ be a strictly $m$-pseudoconvex domain. The statements above are
still valid for continuous $m-\omega$-sh functions $v,w$ and $-1\leq\rho\leq
0$ satisfying $-1\leq v\leq w\leq 0$ and $v=w$ in a neighborhood of
$\partial\Omega$.

In fact, there exist decreasing sequences of $m-\omega$-sh functions
$v_{j},w_{j}$ and $\rho_{j}$ belonging to $C^{\infty}(\overline{\Omega})$ such
that $v_{j}\downarrow v$, $w_{j}\downarrow w$ and $\rho_{j}\downarrow\rho$
(uniformly) in $\overline{\Omega}$, and moreover,

$-1\leq v_{j}\leq w_{j}\leq 0,\quad-1\leq\rho_{j}\leq 0.$

For plurisubharmonic functions the usual convolution with standard kernels
produces the approximating sequence and hence, the property: $v_{j}=w_{j}$
near the boundary is preserved. Then, the integration by parts is not affected
and passing to the limit as $j\to+\infty$ gives the desired inequalities.
However, in this new setting we used a different way to obtain the
approximating sequence. The property that $v_{j}=w_{j}$ near the boundary of
$\Omega$ needs to be verified, which is possible via the stability estimates
for complex Hessian equations. However, we can get around this by showing the
uniform convergence to zero of the sequence $h_{j}=w_{j}-v_{j}$ near the
boundary.

Let $\Omega^{\prime}\subset\subset\Omega$ be a smooth domain such that $v=w$
outside $\Omega^{\prime}$. Let
$T_{j}=(dd^{c}\rho_{j})^{k}\wedge(dd^{c}v_{j})^{s}$, where $k+s\leq m$. Then,
it follows from the weak convergence [KN16, Proposition 2.11] and the CLN
inequality [KN16, Proposition 2.9] that for $p\geq 1$,

$\lim_{j\to\infty}\int_{\Omega^{\prime}}h_{j}^{p}T_{j}\wedge\omega^{n-k-s}=\int_{\Omega}h^{p}T\wedge\omega^{k-s}.$

Therefore, we reduce the required inequality to the case of smooth functions
$v_{j},w_{j}$ and $\rho_{j}$. However the integration by parts in (2.20) will
contain the extra boundary terms:

$\int_{\Omega^{\prime}}h_{j}^{p}dd^{c}\rho_{j}\wedge
T_{j}\wedge\omega^{n-k-s}=\int_{\Omega^{\prime}}\rho_{j}dd^{c}(h_{j}^{p}\omega^{n-k-s})\wedge
T_{j}+E_{1}+E_{2},$

where

$\displaystyle E_{1}$
$\displaystyle=\int_{\partial\Omega^{\prime}}h_{j}^{p}d^{c}\rho_{j}\wedge\omega^{n-k-s}\wedge
T_{j};$ $\displaystyle E_{2}$
$\displaystyle=-\int_{\partial\Omega^{\prime}}\rho_{j}d^{c}(h^{p}_{j}\omega^{n-k-s})\wedge
T_{j}$
$\displaystyle=-\int_{\partial\Omega^{\prime}}\rho_{j}h_{j}^{p-1}(pd^{c}h_{j}\wedge\omega^{n-k-s}+h_{j}d^{c}\omega^{n-k-s})\wedge
T_{j}.$

By the CLN inequality and $h_{j}\to 0$ uniformly on a neighborhood of
$\partial\Omega^{\prime}$ as $j\to\infty$, the two boundary terms go to zero
when we pass to the limit.

######  Remark 2.17.

We will see later that the above statements also hold for bounded
$m-\omega$-sh functions once we define the wedge product for currents related
to such functions and prove the weak convergence under decreasing sequences.

##  3\. Wedge product for bounded functions

In this section we prove the existence of the wedge product of currents where
$dd^{c}$ operator is applied to bounded $m-\omega$-sh functions. Then, we
introduce a weighted Sobolev space associated to these resulted positive
currents. This is a crucial technical tool that allows us to establish the
integration by part formulae for (non-closed) currents of order zero.

###  3.1. The wedge product for bounded $m-\omega$-sh functions

Let $\Omega\subset\mathbb{C}^{n}$ be a bounded open set. We are going to show
that the wedge product of bounded $m-\omega$-sh functions can be defined
inductively by

$dd^{c}v\wedge dd^{c}u=dd^{c}(vdd^{c}u)$

and similarly for more terms. Let us start with the following observation.

######  Lemma 3.1.

Let $u$ be a bounded $m-\omega$-sh function. Then, $dd^{c}u$ is a real
$(1,1)$-current of order zero whose coefficients are singed Radon measures.

###### Proof.

Without loss of generality we may assume $-1\leq u\leq 0$. We will show that
there is a signed measure $\mu$ such that

$dd^{c}u(\phi)=\int\phi d\mu$

for every (smooth) test $(n-1,n-1)$-form. By a theorem in distribution theory
in Federer’s book [Fe69, Section 4.15] it is enough to show that for all
$K\subset\Omega$ compact there exists $C=C(K,\Omega)$ such that

$|dd^{c}u(\phi)|\leq C\|\phi\|_{K}$

for any smooth test form $\phi$ with $\mbox{supp }\phi\subset K$. Here we
denote

$\|\phi\|_{K}=\sum_{i,j}\sup_{K}|\phi_{i\bar{j}}|,$

where $\phi_{i\bar{j}}$’s are coefficients of the form $\phi$. By the
definition of $dd^{c}$-operator for currents

$dd^{c}u(\phi)=u(dd^{c}\phi)=\int u\wedge dd^{c}\phi.$

Thus, for a smooth sequence of $m-\omega$-sh functions $\\{u_{s}\\}$ that
decrease to $u$ we have

$dd^{c}u(\phi)=\lim_{s\to\infty}(dd^{c}u_{s})(\phi)=\lim_{s\to\infty}\int
dd^{c}u_{s}\wedge\phi.$

Since $dd^{c}u_{s}\in\Gamma_{m}(\omega)$, it follows from [KN16, Corollary
2.4] that

(3.1) $|dd^{c}u_{s}\wedge\phi|\leq
C\|\phi\|_{K}\;dd^{c}u_{s}\wedge\omega^{n-1}$

for some universal constant depending on $n,m$. It follows that

$\left|\int dd^{c}u_{s}\wedge\phi\right|\leq
C\|\phi\|_{K}\int_{K}dd^{c}u_{s}\wedge\omega^{n-1}.$

Thanks to the CLN inequality [KN16, Proposition 2.9] we know that the last
integral on the right hand side is bounded by a constant
$C(K,\Omega)\|u_{s}\|_{L^{\infty}}$. This finishes the proof.

Furthermore, for any test function $\chi\geq 0$,

$0\leq\lim_{s\to\infty}\int\chi\omega^{n-1}\wedge
dd^{c}u_{s}=dd^{c}u(\chi\omega^{n-1}).$

In other words, $dd^{c}u\wedge\omega^{n-1}$ is a (positive) Radon measure.
Moreover, it follows from (3.1) that the total variation of coefficients of
$dd^{c}u$ is dominated by $dd^{c}u\wedge\omega^{n-1}$. Therefore, the
coefficients of $T=dd^{c}u$ are signed Radon measures. ∎

It follows from Lemma 3.1 that we can multiply $dd^{c}u$ by a bounded function
$v$. In fact, we can continue to define inductively the wedge product in this
way and obtain a real current of order zero for up to $(m-1)$ bounded
functions.

######  Lemma 3.2.

Let $u_{1},...,u_{p}$ with $1\leq p\leq m-1$ be bounded $m-\omega$-sh
functions. Assume $T_{0}=1$ and $T_{p-1}=dd^{c}u_{p-1}\wedge\cdots\wedge
dd^{c}u_{1}$. Then, the current

$dd^{c}u_{p}\wedge\cdots\wedge dd^{c}u_{1}:=dd^{c}[u_{p}T_{p-1}]$

is a well-defined real $(p,p)$-current of order zero whose coefficients are
signed Radon measures. Moreover,

$\mathscr{L}_{p}(u_{p},...,u_{1}):=dd^{c}u_{p}\wedge\cdots\wedge
dd^{c}u_{1}\wedge\omega^{n-m}$

is a positive $(n-m+p,n-m+p)$-current.

###### Proof.

We argue by induction in $p$. If $p=1$, then this is Lemma 3.1. Assume that it
holds for $1\leq p\leq m-2$, we need to show that the lemma holds for $p+1$.
For the simplicity of notation we present the argument for $p+1=2$ and note
that the proof in the general case is completely the same. Let us write
$u_{1}=u$ and $u_{2}=v$ and assume that $-1\leq u,v\leq 0$. Since $dd^{c}u$
has signed measure coefficients, the currents

$S=vdd^{c}u,\quad\text{and }dd^{c}v\wedge dd^{c}u:=dd^{c}S=dd^{c}(vdd^{c}u).$

are well-defined. Our next goal is to show that the real (closed) current
$dd^{c}S$ of bidegree $(2,2)$ has order zero, thus its coefficients are signed
Radon measures.

As above we will show that for each compact set $K\subset\Omega$, there is a
uniform constant $C$ such that

(3.2) $|dd^{c}S(\phi)|\leq C\|\phi\|_{K}$

for every smooth test form with $\mbox{supp }\phi\subset K$, where
$\|\phi\|_{K}$ denotes the $C^{0}$-norm the form $\phi$. To this end, let
$\\{v_{\ell}\\}_{\ell\geq 1}$ and $\\{u_{s}\\}_{s\geq 1}$ be sequences of
smooth $m-\omega$-sh functions such that

$v_{\ell}\downarrow v,\quad\text{and}\quad u_{s}\downarrow u.$

We may also assume that $-1\leq u_{s},v_{\ell}\leq 0$. Fix $\phi$ a test form.
By definition we have

(3.3) $dd^{c}S(\phi)=S(dd^{c}\phi)=\int vdd^{c}u\wedge dd^{c}\phi.$

Since $dd^{c}u\wedge dd^{c}\phi$ has signed Radon measure coefficients, it
follows from the dominated convergence theorem that

$\int vdd^{c}u\wedge dd^{c}\phi=\lim_{\ell\to\infty}\int v_{\ell}dd^{c}u\wedge
dd^{c}\phi=\lim_{\ell\to\infty}\lim_{s\to\infty}\int v_{\ell}dd^{c}u_{s}\wedge
dd^{c}\phi.$

By the integration by parts formula,

$\int v_{\ell}dd^{c}u_{s}\wedge dd^{c}\phi=\int dd^{c}v_{\ell}\wedge
dd^{c}u_{s}\wedge\phi.$

Notice that $m-1\geq 2$. Applying [KN16, Corollary 2.4] for
$\gamma_{1}=dd^{c}u_{s},\gamma_{2}=dd^{c}v_{\ell}$ belonging to
$\Gamma_{m}(\omega)$, we get

(3.4) $\left|dd^{c}v_{\ell}\wedge dd^{c}u_{s}\wedge\phi\right|\leq
C\|\phi\|_{K}[dd^{c}(u_{s}+v_{\ell})]^{2}\wedge\omega^{n-2}.$

Integrating both sides and then applying the CLN inequality for the right hand
side we get

$\left|\int dd^{c}v_{\ell}\wedge dd^{c}u_{s}\wedge\phi\right|\leq
C(K,\Omega)\|\phi\|_{K}.$

Letting $s\to\infty$ and then $\ell\to\infty$, in this order, we conclude from
(3.2) the desired estimate (3.3). Hence, $dd^{c}S$ is a $(2,2)$ current of
order zero.

Furthermore, for a test function $\chi\geq 0$,

$dd^{c}S\wedge\omega^{n-2}(\chi)=\lim_{\ell\to\infty}\lim_{s\to\infty}\int\chi
dd^{c}v_{\ell}\wedge dd^{c}u_{s}\wedge\omega^{n-2}\geq 0.$

Thus, $dd^{c}S\wedge\omega^{n-2}$ is a positive Radon measure. It follows from
(3.4) (by passing to the limit) that the coefficients of $dd^{c}S$ are signed
Radon measures.

Finally, for smooth $m-\omega$-sh functions $u_{1}^{j_{1}},...,u_{p}^{j_{p}}$,
then the form

$\mathscr{L}_{p}(u_{p}^{j_{p}},...,u_{1}^{j_{1}})=dd^{c}u_{p}^{j_{p}}\wedge\cdots\wedge
dd^{c}u_{1}^{j_{1}}\wedge\omega^{n-m}$

is positive by Gårding’s inequality [Ga59]. Letting $j_{1}\to\infty$,
…$j_{p}\to\infty$, in this order, for decreasing sequences approximating
$u_{p},...,u_{1}$ we get that $\mathscr{L}_{p}(u_{p},...,u_{1})$ is a positive
current of bidegree $(n-m+p,n-m+p)$. ∎

Thus, we defined inductively for bounded $m-\omega$-sh functions,
$u_{1},...,u_{m-1}$, a real closed current

(3.5) $S:=dd^{c}u_{m-1}\wedge\cdots\wedge dd^{c}u_{1}$

of order zero whose coefficients are signed Radon measures. Furthermore,

(3.6) $S\wedge\omega^{n-m+1}$

is a positive Radon measure. In the last step we multiply $S$ by a bounded
function $v=u_{m}$.

######  Theorem 3.3.

Let $S$ be as in (3.5) and let $v$ be a bounded $m-\omega$-sh function. Then,

$T:=dd^{c}(vS)$

is a real closed $(m,m)$-current and $T\wedge\omega^{n-m}$ is a positive Radon
measure.

###### Proof.

Let $\chi\geq 0$ be a test function. By definition

$T\wedge\omega^{n-m}(\chi)=T(\chi\omega^{n-m})=\int vS\wedge
dd^{c}(\chi\omega^{n-m}).$

Since $S$ is of order zero, it follows from the dominated convergence theorem
that for a sequence of smooth $m-\omega$-sh functions $v_{\ell}\downarrow v$,

$\int vS\wedge dd^{c}(\chi\omega^{n-m})=\lim_{\ell\to\infty}\int
v_{\ell}S\wedge dd^{c}(\chi\omega^{n-m}).$

Using the inductive definition of $S$ and the integration by parts, we get

$\displaystyle\int v_{\ell}S\wedge dd^{c}(\chi\omega^{n-m})$
$\displaystyle=\lim_{j_{m-1}\to\infty}\cdots\lim_{j_{1}\to\infty}\int
dd^{c}v_{\ell}\wedge dd^{c}u^{j_{m-1}}\wedge\cdots\wedge u^{j_{1}}\wedge
dd^{c}(\chi\omega^{n-m})$
$\displaystyle=\lim_{j_{m-1}\to\infty}\cdots\lim_{j_{1}\to\infty}\int
dd^{c}v_{\ell}\wedge dd^{c}u^{j_{m-1}}\wedge\cdots\wedge
dd^{c}u^{j_{1}}\wedge(\chi\omega^{n-m}).$

Since $v_{\ell},u^{j_{1}},...,u^{j_{m-1}}$ are smooth $m-\omega$-sh functions,
the last integrand is positive and the proof follows. ∎

Consequently the following definition is justified.

######  Definition 3.4.

Let $u_{1},...,u_{p}$ with $1\leq p\leq m$ be bounded $m-\omega$-sh functions
in $\Omega$. The wedge product is given inductively by

$dd^{c}u_{p}\wedge\cdots\wedge dd^{c}u_{1}:=dd^{c}[u_{p}T_{p-1}],$

where $T_{0}=1$ and $T_{p-1}=dd^{c}u_{p-1}\wedge\cdots\wedge dd^{c}u_{1}$.

We are most interested in the following of positive Radon measures.

######  Definition 3.5.

Let $u$ be a bounded $m-\omega$-sh function. Then, the Hessian operator
$H_{m}(u)$ is defined by

$H_{m}(u):=(dd^{c}u)^{m}\wedge\omega^{n-m}=\mathscr{L}_{m}(u,...,u).$

Moreover, for $1\leq s\leq m$, we also write
$H_{s}(u)=(dd^{c}u)^{s}\wedge\omega^{n-s}$.

For convenience, we summarize the results in Lemma 3.2 and Theorem 3.3 as
follows. Let $\\{u_{s}^{j}\\}_{j\geq 1}$ be decreasing sequences such that
$u_{s}^{j}\downarrow u_{s}$ for each $1\leq s\leq m$. Define

(3.7) $T^{j_{p}\cdots j_{1}}=dd^{c}u_{p}^{j_{p}}\wedge\cdots\wedge
dd^{c}u_{1}^{j_{1}}.$

Then, we have $T^{j_{p}\cdots j_{2}}\wedge
dd^{c}u_{1}=\lim_{j_{1}\to\infty}T^{j_{p}\cdots j_{1}}$, and consequently

(3.8) $dd^{c}u_{p}\wedge\cdots\wedge
dd^{c}u_{1}=\lim_{j_{p}\to\infty}\cdots\lim_{j_{1}\to\infty}T^{j_{p}\cdots
j_{1}}.$

For $1\leq p\leq m-1$, each limit in (3.8) is in the sense of currents of
order zero. For $p=m$, we have

(3.9) $dd^{c}u_{p}\wedge\cdots\wedge
dd^{c}u_{1}\wedge\omega^{n-m}=\lim_{j_{p}\to\infty}\cdots\lim_{j_{1}\to\infty}T^{j_{p}\cdots
j_{1}}\wedge\omega^{n-m},$

where each limit is also in the sense of currents of order zero.

######  Remark 3.6.

At this point the definition of the wedge product in Definition 3.4 is not
symmetric with respect to $u_{1},...,u_{p}$. Furthermore, if the metric
$\omega$ is Kähler or just the standard Kähler form $\beta=dd^{c}|z|^{2}$ in
$\mathbb{C}^{n}$, then by (3.9) it follows that Definition 3.4 coincides with
the ones given before in [Bl05], [DK14].

By taking the limit inductively as in (3.8) and (3.9) we can state now the CLN
inequality for bounded functions.

######  Proposition 3.7 (CLN inequality).

Let $K\subset\subset U\subset\subset\Omega$, where $K$ is compact and $U$ is
open. Let $u,u_{1},...,u_{p}$, be bounded $m-\omega$-sh functions in $\Omega$,
where $1\leq p\leq m$. Then, there exists a constant $C$ depending on
$K,U,\omega$ such that

  * (a)

$\int_{K}(dd^{c}u)^{p}\wedge\omega^{n-p}\leq C(1+\|u\|_{L^{\infty}(U)})^{p}$;

  * (b)

$\int_{K}dd^{c}u_{1}\wedge\cdots\wedge dd^{c}u_{p}\wedge\omega^{n-p}\leq
C\left(1+\sum_{s=1}^{p}\|u_{s}\|_{L^{\infty}(U)}\right)^{p}.$

###### Proof.

(a) Without loss of generality we may assume that
$\|u\|_{L^{\infty}(\Omega)}\leq 1$. We can cover $K$ by finitely many small
balls, hence by localization principle we can assume that $K$ and $U$ are
concentric balls. Let $\\{u^{j}\\}_{j\geq 1}$ be sequences of smooth
$m-\omega$-sh functions such that $u^{j}\downarrow u$ as $j\to\infty$. Let
$0\leq\chi\leq 1$ be a cut-off function such that $\chi\equiv 1$ on $K$ and
$\mbox{supp }\chi\subset U$. By the CLN inequality for smooth functions [KN16,
Proposition 2.9]

$\int\chi\mathscr{L}_{p}(u^{j_{p}},...,u^{j_{1}})\wedge\omega^{m-p}\leq
C\left(1+\|\sum_{s=1}^{p}u^{j_{s}}\|_{L^{\infty}(U)}^{p}\right).$

By the Hartogs lemma for $\omega$-sh functions [GN18, Lemma 9.14] it follows
that

$\lim_{j\to\infty}\|1+u^{j}\|_{L^{\infty}(U)}=\lim_{j\to\infty}\sup_{\overline{U}}(1+u^{j})=1+\sup_{\overline{U}}u\leq
1+\|u\|_{L^{\infty}(U)}.$

Let $j_{1}\to\infty$, ….,$j_{p}\to\infty$, in this order we get

$\displaystyle\int_{K}\mathscr{L}_{p}(u)\wedge\omega^{m-p}$
$\displaystyle\leq\lim_{j_{p}\to\infty}\cdots\lim_{j_{1}\to\infty}\int\chi\mathscr{L}_{p}(u^{j_{p}},...,u^{j_{1}})\wedge\omega^{m-p}$
$\displaystyle\leq
C\lim_{j_{p}\to\infty}\cdots\lim_{j_{1}\to\infty}\left(1+\|\sum_{s=1}^{p}u^{j_{s}}\|_{L^{\infty}}\right)^{p}$
$\displaystyle=C(1+\|u\|_{L^{\infty}(U)})^{p}.$

(b) Observe that for $v:=u_{1}+\cdots+u_{p}$ we have
$\mathscr{L}_{p}(v)\geq\mathscr{L}(u_{1},...,u_{p})$ as positive currents. So,
(b) is an immediate consequence of (a). ∎

We also need a simplified version of Lemma 2.3 for particular positive
currents. Let $-1\leq u_{1},...,u_{m-2}\leq 0$ be bounded $m-\omega$-sh
functions in $\Omega$. Denote by $T$ the current

(3.10) $T=dd^{c}u_{1}\wedge\cdots\wedge dd^{c}u_{m-2}\wedge\omega^{n-m}.$

######  Corollary 3.8.

Let $\phi$ be a Borel function such that $\mbox{supp
}\phi=K\subset\subset\Omega$. Assume also $|\phi|\leq 1$. Let $0\leq h\leq 1$
be a smooth $m-\omega$ function in $\Omega$. There exists a constant
$C=C(\omega,K)$ such that

$\left|\int\phi\;dh\wedge d^{c}\omega\wedge T\right|^{2}\leq
C\int|\phi|\;T\wedge\omega^{2}.$

###### Proof.

By Lemma 2.3, it is enough to show that $\int|\phi|dh\wedge d^{c}h\wedge
T\wedge\omega\leq C.$ In fact, since $h^{2}$ and $h$ are $m-\omega$-sh, we
have

$2dh\wedge d^{c}h\wedge T\wedge\omega=[dd^{c}h^{2}-2hdd^{c}h]\wedge
T\wedge\omega\leq dd^{c}h^{2}\wedge T\wedge\omega.$

Multiplying both sides by $|\phi|$ and integrating them, we get

$\int|\phi|dh\wedge d^{c}h\wedge T\wedge\omega\leq\int|\phi|dd^{c}h^{2}\wedge
T\wedge\omega.$

Using the assumption on $\phi$ the right hand side is bound by

$\int_{K}dd^{c}h^{2}\wedge T\wedge\omega.$

The desired estimate follows from the CLN inequality for bounded functions
(Proposition 3.7). ∎

###  3.2. Weighted Sobolev space

Let $K\subset\Omega$ be a compact subset and let $\psi$ be a strictly
plurisubharmonic defining function for $\Omega$. Let us denote by
$P^{*}=P^{*}(\Omega,K,\psi)$ the set of bounded $m-\omega$-sh functions that

$u(z)=\psi,\quad z\in\Omega\setminus K.$

Let $u_{1},..,u_{p}\in P^{*}$ with $1\leq p\leq m-1$ and denote

(3.11) $\tau=dd^{c}u_{1}\wedge\cdots\wedge dd^{c}u_{p},$

and conventionally $\tau=1$ for $p=0$. By definition this is a closed
$(p,p)$-current of order zero. We first make an additional assumption on the
current $\tau$. By considering $u_{i}=u_{i}+\delta|z|^{2}$ for some $\delta>0$
small, we may assume that there is $c_{0}>0$ such that

(3.12) $\tau\wedge\omega^{n-p-1}\geq c_{0}\omega^{n-1}.$

Hence, the trace measure (with respect to $\omega$) of the positive current
$\tau\wedge\omega^{n-p}$ satisfies

(3.13) $\mu_{p}:=\tau\wedge\omega^{n-p}\geq c_{0}\omega^{n}.$

Let us introduce a norm associated to $\tau$ on the space smooth functions
$C^{\infty}(\Omega,\mathbb{R})$. If $v,w$ are smooth functions in $\Omega$,
then we can define

$\left<v,w\right>_{\tau}=\int dv\wedge d^{c}w\wedge\tau\wedge\omega^{n-p-1},$

and

$\|v\|_{\tau}^{2}=\left<v,v\right>_{\tau}=\int dv\wedge
d^{c}v\wedge\tau\wedge\omega^{n-p}.$

Let us consider the following inner product

$\left<v,w\right>=\int vw\;\beta^{n}+\left<v,w\right>_{\tau},$

where $\beta=dd^{c}|z|^{2}$, and the associated norm

$\|v\|^{2}=\left<v,v\right>=\|v\|_{L^{2}}^{2}+\|v\|_{\tau}^{2}.$

The assumption makes this inner product positive definite. In fact, it is
clear that $\left<\bullet,\bullet\right>$ satisfies the properties of an inner
product, namely

  * (i)

$\left<v,v\right>\geq 0$ and $\left<v,v\right>=0$ if and only if $v=0$.

  * (ii)

$\left<av_{1}+bv_{2},w\right>=a\left<v_{1},w\right>+b\left<v_{2},w\right>$.

  * (iii)

$\left<v,w\right>=\left<w,v\right>$.

This implies also that it satisfies the triangle inequality

$\left|\left<v,w\right>\right|\leq\sqrt{\left<v,v\right>\left<w,w\right>}.$

Let us denote by $W^{1,2}(\Omega,\mu_{p})$ the real Hilbert space completion
of $C^{\infty}(\Omega,\mathbb{R})$ with respect to this norm. Let us state a
property of this space immediately following from the definition.

######  Lemma 3.9.

$W^{1,2}(\Omega,\mu_{p})$ is uniformly convex. Consequently, it is reflexive.

Our goal is to show that for each $0\leq p\leq m-1$,

$P^{*}\subset W^{1,2}(\Omega,\mu_{p}).$

If we take $u_{p}=\psi$ the strictly psh defining function for $\Omega$, then
clearly

$W^{1,2}(\Omega,\mu_{p})\subset W^{1,2}(\Omega,\mu_{p-1}).$

Therefore, we only need to prove two special cases for $p=0$ and $p=m-1$. Let
us start with this simpler case $p=0$, i.e., $\tau=1$,
$\tau\wedge\omega^{n-1}=\omega^{n-1}$ and $\mu_{0}=\omega^{n}$ which is
equivalent to the Lebesgue measure $dV=\beta^{n}$. This is a classical result
if $\omega=\beta$ is the standard Käher metric (see e.g. [Hor94, Proposition
3.4.19]).

######  Lemma 3.10.

All functions in $P^{*}$ belong to the Sobolev space $W^{1,2}(\Omega)$.

###### Proof.

Let $u\in P^{*}$. By the approximation theorem there is $\\{u_{j}\\}_{j\geq
1}\subset P^{*}$ a sequence of smooth functions such that $u_{j}\downarrow u$
pointwise. We may assume that $0\leq u,u_{j}\leq 1$. Observe that the sequence
$\\{u_{j}\\}_{j\geq 1}$ is uniformly bounded in $W^{1,2}(\Omega)$. Indeed, by
the $\omega$-subharmonicity

$\displaystyle\int du_{j}\wedge d^{c}u_{j}\wedge\omega^{n-1}$
$\displaystyle=\int\left(\frac{1}{2}dd^{c}u_{j}^{2}-2u_{j}dd^{c}u_{j}\right)\wedge\omega^{n-1}$
$\displaystyle\leq\frac{1}{2}\int dd^{c}u_{j}^{2}\wedge\omega^{n-1}.$

The last integral is bounded by a constant $C=C(K,\psi,\Omega)$ by an
application of the CLN inequality. Therefore, the sequence of (vector-valued)
functions $\nabla u_{j}$ is uniformly bounded in $L^{2}(\Omega)$. Thus, by the
weak compactness theorem, $\nabla u_{j}$ converges weakly to a vector-valued
function ${\bf v}\in L^{2}(\Omega)$.

To complete the proof we are going to show that ${\bf v}=\nabla u$. First,
note that $u_{j}\to u$ in $L^{2}(\Omega)$ by the dominated convergence
theorem. So, $(u_{j},\nabla u_{j})\to(u,{\bf v})$ weakly in the Hilbert space
$L^{2}(\Omega)\times L^{2}(\Omega,\mathbb{R}^{n})$. The Mazur lemma [Yo80,
Chapter V, Theorem 2] implies that there is a sequence of convex combinations

$\widetilde{u}_{\ell}=\sum_{j=1}^{\ell}\lambda_{\ell,j}(u_{j},\nabla
u_{j}),\quad\lambda_{\ell,j}\geq 0\text{ and
}\sum_{j=1}^{\ell}\lambda_{\ell,j}=1,$

which converges in norm to $(u,{\bf v})$ in $L^{2}(\Omega)\times
L^{2}(\Omega,\mathbb{R}^{n})$. It follows that

$w_{\ell}=\sum_{j=1}^{\ell}\lambda_{\ell,j}u_{j}$

is a Cauchy sequence in $W^{1,2}(\Omega)$. Hence, there exist a limit function

$\widetilde{u}=\lim_{\ell\to\infty}w_{\ell}\in W^{1,2}(\Omega).$

It follows that $u=\widetilde{u}$ and ${\bf v}=\nabla\widetilde{u}=\nabla u$.
∎

We proceed to prove the most general case $1\leq p=m-1$ when

$\mu=\mu_{p}:=\tau\wedge\omega^{n-m+1}$

is a positive Radon measure. Equivalently, we wish to show that if $v\in
P^{*}$, then the form $dv$ has coefficients in $L^{2}(\Omega,d\mu)$. Note that
for smooth function $v$ we have

$\displaystyle\int|\nabla v|^{2}dV\leq\frac{1}{c_{0}}\int dv\wedge
d^{c}v\wedge\tau\wedge\omega^{n-m}.$

It implies that $W^{1,2}(\Omega,\mu)\subset W^{1,2}(\Omega)$. In particular,
the gradient $\nabla u$ is well-defined in the space $W^{1,2}(\Omega,\mu)$.

######  Proposition 3.11.

Let $v\in P^{*}$ and $\\{v_{\ell}\\}_{\ell\geq 0}\subset P^{*}$ be a sequence
of smooth $m-\omega$-sh functions such that $v_{\ell}\downarrow v$ point-wise
in $\Omega$. Then $v_{\ell}$ converges weakly to $v$ in $W^{1,2}(\Omega,\mu)$,
i.e., for every $w\in W^{1,2}(\Omega,\mu)$,

$\lim_{\ell\to\infty}\left<v_{\ell},w\right>=\left<v,w\right>.$

In particular, $P^{*}\subset W^{1,2}(\Omega,\mu)$.

###### Proof.

We may assume that $0\leq v\leq v_{\ell}\leq 1$. Then, there exists a uniform
constant $C=C(\omega,n,m,\psi,\Omega)$ such that

$\|\nabla v_{\ell}\|^{2}_{\tau}=\int dv_{\ell}\wedge
d^{c}v_{\ell}\wedge\tau\wedge\omega^{n-m}\leq C\|v_{\ell}\|_{L^{\infty}}^{2}.$

In fact, since $dv_{\ell}\wedge
d^{c}v_{\ell}=\frac{1}{2}dd^{c}v_{\ell}^{2}-v_{\ell}dd^{c}v_{\ell},$ and
$dd^{c}v_{\ell}\wedge\tau\wedge\omega^{n-m}$ is a positive measure and
$v_{\ell}\geq 0$, it follows that

$\int dv_{\ell}\wedge d^{c}v_{\ell}\wedge\tau\wedge\omega^{n-m}\leq\int
dd^{c}v_{\ell}^{2}\wedge\tau\wedge\omega^{n-m}.$

Since $v_{\ell}^{2}$ is also $m-\omega$-sh, the desired inequality follows
from the CLN inequality.

In the same way as in the previous proof, via Mazur’s lemma, one proves that
$v\in W^{1,2}(\Omega,\mu)$. ∎

######  Remark 3.12.

The proofs given in Lemma 3.10 and Proposition 3.11 are inspired by the one in
the book by Heinonen, Kilpeläinen and Martio [HKM06, Theorem 1.30].

###  3.3. Integration by parts for bounded functions

In this section we show ”integration by parts inequalities” for bounded
$m-\omega$-sh functions which are smooth near the boundary. In particular, it
holds for functions in the class $P^{*}$. First we deal with the case of the
wedge product of $(m-2)$ bounded functions and consider

(3.14) $\eta=dd^{c}u_{1}\wedge\cdots\wedge dd^{c}u_{m-2}$

where $u_{1},...,u_{m-2}\in P^{*}$ as in Section 3.2. Recall that we also
impose the assumption (3.12) for $\eta$, namely

(3.15) $\eta\wedge\omega^{n-m+1}\geq c_{0}\omega^{n-1},$

and denote the corresponding trace measure by $\mu$:

(3.16) $\mu=\eta\wedge\omega^{n-m+2}.$

The following integration by parts formula for currents of of order zero is
used frequently below. Let $\phi$ and $\rho$ be smooth functions such that
$\phi=0$ near $\partial\Omega$. Then,

(3.17) $\displaystyle\int\phi\;dd^{c}\rho\wedge\eta\wedge\omega^{n-m+1}$
$\displaystyle=-\int d\phi\wedge d^{c}\rho\wedge\eta\wedge\omega^{n-m+1}$
$\displaystyle\quad+\int\phi d^{c}\rho\wedge
d\omega\wedge\eta\wedge\omega^{n-m}.$

If $v,w\in P^{*}\subset W^{1,2}(\Omega,\mu)$ are smooth, then

$\left<v,w\right>_{\eta}=\int dv\wedge d^{c}w\wedge\eta\wedge\omega^{n-m+1}.$

By abusing the notation for general $v,w\in P^{*}$ we will write the integral
on the right understanding that it is the inner product
$\left<v,w\right>_{\eta}$. The first version of the ”integration by part
inequality” in the class $P^{*}$ reads as follows.

######  Lemma 3.13.

Let $\\{v_{\ell}\\}_{\ell\geq 1}$ be a sequence of smooth functions from
$P^{*}$ and let $w\in P^{*}$ be smooth. Assume $v_{\ell}\leq w$ and
$v_{\ell}\downarrow v\in P^{*}$. Then,

$\displaystyle-\int d(w-v)\wedge d^{c}v\wedge\eta\wedge\omega^{n-m+1}$
$\displaystyle\leq\int(w-v)dd^{c}v\wedge\eta\wedge\omega^{n-m+1}$
$\displaystyle+C\left(\int(w-v)\eta\wedge\omega^{n-m+2}\right)^{\frac{1}{2}}.$

The constant $C$ depends on $\omega,K,\psi$ and the uniform norm of functions,
but it is independent of $\delta$ in (3.12).

###### Proof.

Without loss of generality we may assume that $0\leq v,v_{\ell},w\leq 1$ and
$0\leq u_{i}\leq 1$. By the weak convergence in the space
$W^{1,2}(\Omega,\mu)$ (Propostion 3.11) and the dominated convergence theorem
it is enough to prove that

$\displaystyle-\int d(w-v_{\ell})\wedge d^{c}v\wedge\eta\wedge\omega^{n-m+1}$
$\displaystyle\leq\int(w-v_{\ell})dd^{c}v\wedge\eta\wedge\omega^{n-m+1}$
$\displaystyle+C\left(\int(w-v_{\ell})\eta\wedge\omega^{n-m+2}\right)^{\frac{1}{2}}.$

To this end we use once more the weak convergence property when $\ell$ is
fixed. Note that $w-v_{\ell}$ is a continuous function whose support is
compact in $\Omega$. It follows from the weak convergence of currents (of
order zero) that

$\int(w-v_{\ell})dd^{c}v\wedge\eta\wedge\omega^{n-m+1}=\lim_{j\to\infty}\int(w-v_{\ell})dd^{c}v_{j}\wedge\eta\wedge\omega^{n-m+1}$

Now for each $\ell$ and $j$, the integration by parts formula (3.17) gives

$\displaystyle-\int d(w-v_{\ell})\wedge
d^{c}v_{j}\wedge\eta\wedge\omega^{n-m+1}$
$\displaystyle=\int(w-v_{\ell})dd^{c}v_{j}\wedge\eta\wedge\omega^{n-m+1}$
$\displaystyle\quad+\int(w-v_{\ell})d^{c}v_{j}\wedge
d\omega\wedge\eta\wedge\omega^{n-m}.$

By Corollary 3.8 applied for $\phi=w-v_{\ell}$ and CLN inequality (Proposition
3.7) for bounded functions, the second integral on the right hand side is
bounded by

$C\left(\int(w-v_{\ell})\eta\wedge\omega^{n-m+2}\right)^{\frac{1}{2}}.$

The desired inequality follows by letting $j\to\infty$. ∎

######  Lemma 3.14.

Let $v,w$ and $\\{v_{\ell}\\}$ be as in Lemma 3.13. Let $\rho$ be a bounded
$m-\omega$-sh in $\Omega$ such that $-1\leq\rho\leq 0$. Then,

$\displaystyle\int_{\Omega}(w-v)dd^{c}\rho\wedge\eta\wedge\omega^{n-m+1}$
$\displaystyle\leq-\int_{\Omega}d(w-v)\wedge
d^{c}\rho\wedge\eta\wedge\omega^{n-m+1}$
$\displaystyle\quad+C\left(\int_{\Omega}(w-v)\wedge\eta\wedge\omega^{n-m+2}\right)^{\frac{1}{2}}.$

The dependence of $C$ is the same as in the previous lemma.

###### Proof.

Since the supports of the integrands on both sides are contained in $K$, we
replace $\rho$ by $\max\\{\rho,A\psi\\}$ for some $A>0$ large such that
$A\psi\leq-1$ on a neighborhood of $K$. Thus, we may assume that $\rho\in
P^{*}(\Omega,K^{\prime},A\psi)$ for a compact subset $K\subset
K^{\prime}\subset\subset\Omega$. By dominated convergence theorem for Radon
measures and the weak convergence in Proposition 3.11 it is enough to prove
the inequality for $v_{\ell}$ in the place of $v$, that is

(3.18)
$\displaystyle\int_{\Omega}(w-v_{\ell})dd^{c}\rho\wedge\eta\wedge\omega^{n-m+1}$
$\displaystyle\leq-\int_{\Omega}d(w-v_{\ell})\wedge
d^{c}\rho\wedge\eta\wedge\omega^{n-m+1}$
$\displaystyle\quad+C\left(\int_{\Omega}(w-v_{\ell})\eta\wedge\omega^{n-m+2}\right)^{\frac{1}{2}}.$

For a fixed $\ell$ the function $w-v_{\ell}$ is continuous, with its support
in $\Omega$. We can apply the limit argument once more to assume that $\rho$
also is smooth as follows. By the approximation theorem for $m-\omega$-sh
functions we can find a decreasing sequence $\\{\rho_{j}\\}_{j\geq 1}\subset
P^{*}$ of smooth $m-\omega$-sh functions such that $\rho_{j}\downarrow\rho$.
We may assume that $-A\leq\rho_{j}\leq 0$. Then,

$dd^{c}\rho_{j}\wedge\eta\wedge\omega^{n-m+1}\to
dd^{c}\rho\wedge\eta\wedge\omega^{n-m+1}$

weakly as measures. So, if the inequality holds for $\rho_{j}$, then so it
does for $\rho$.

Using again the integration by parts formula (3.17) we have

$\displaystyle\int_{\Omega}(w-v_{\ell})dd^{c}\rho_{j}\wedge\eta\wedge\omega^{n-m+1}$
$\displaystyle=-\int_{\Omega}d(w-v_{\ell})\wedge
d^{c}\rho_{j}\wedge\eta\wedge\omega^{n-m+1}$
$\displaystyle\quad+\int_{\Omega}(w-v_{\ell})d^{c}\rho_{j}\wedge
d\omega\wedge\eta\wedge\omega^{n-m}.$

As in the proof of Lemma 3.13, by an application of Cauchy-Schwarz inequality
(Corollary 3.8) and then the CLN inequality for the second integral on the
right hand side, we conclude that the inequality holds for $\rho_{j}$ and so
does (3.18). ∎

Lastly we can state the important estimate for functions in $P^{*}$ and for
$\eta=dd^{c}u_{1}\wedge\cdots\wedge dd^{c}u_{m-2}$, where $u_{1},..,u_{m-2}\in
P^{*}$ without strict positive assumption on $\eta$.

######  Lemma 3.15.

Let $v,w$ and $\\{v_{\ell}\\}$ be as in Lemma 3.13. Let $\rho$ be a bounded
$m-\omega$-sh in $\Omega$ such that $-1\leq\rho\leq 0$. Then,

$\displaystyle\int_{\Omega}(w-v)dd^{c}\rho\wedge\eta\wedge\omega^{n-m+1}$
$\displaystyle\leq
C\left(\int_{\Omega}(w-v)dd^{c}v\wedge\eta\wedge\omega^{n-m+1}\right)^{\frac{1}{2}}$
$\displaystyle\quad+C\left(\int_{\Omega}(w-v)\eta\wedge\omega^{n-m+2}\right)^{\frac{1}{2}}$
$\displaystyle\quad+C\left(\int_{\Omega}(w-v)\eta\wedge\omega^{n-m+2}\right)^{\frac{1}{4}},$

with the same dependence of $C$ as in previous lemmata.

###### Proof.

Observe that it is enough to prove the inequality for
$u_{i}:=u_{i}+\delta|z|^{2}$ and then let $\delta\to 0^{+}$. Therefore without
loss of generality we assume that $\eta$ satisfies conditions (3.15) and
(3.16). As in the proof of Lemma 3.14 we may assume that $\rho\in P^{*}$ and
it is smooth. It follows from (3.18) that

(3.19)
$\displaystyle\int_{\Omega}(w-v_{\ell})dd^{c}\rho\wedge\eta\wedge\omega^{n-m+1}$
$\displaystyle\leq-\int_{\Omega}d(w-v_{\ell})\wedge
d^{c}\rho\wedge\eta\wedge\omega^{n-m+1}$
$\displaystyle\quad+C\left(\int_{\Omega}(w-v_{\ell})\eta\wedge\omega^{n-m+2}\right)^{\frac{1}{2}}.$

It remains to bound the first integral on the right hand side. Using the
Cauchy-Schwarz inequality (Lemma 2.3) for $T=\eta\wedge\omega^{n-m+1}$ we get

(3.20) $\displaystyle\left|\int_{\Omega}d(w-v_{\ell})\wedge d^{c}\rho\wedge
T\right|^{2}$ $\displaystyle\leq C_{1}\int_{\Omega}d(w-v_{\ell})\wedge
d^{c}(w-v_{\ell})\wedge T,$

where $C_{1}$ is the bound for

$C\int_{K}d\rho\wedge d^{c}\rho\wedge T$

and so the dependence of $C_{1}$ is as in the statement which is a simple
consequence of the CLN inequality.

Next, we have

(3.21) $\displaystyle\int_{\Omega}d(w-v_{\ell})\wedge d^{c}(w-v_{\ell})\wedge
T$ $\displaystyle=-\int_{\Omega}(w-v_{\ell})dd^{c}(w-v_{\ell})\wedge T$
$\displaystyle\quad+\int_{\Omega}(w-v_{\ell})d^{c}(w-v_{\ell})\wedge
d\omega\wedge\eta\wedge\omega^{n-m}.$

Let us estimate the last integral. The Cauchy-Schwarz inequality and the CLN
inequality imply that

(3.22) $\displaystyle\left|\int_{\Omega}(w-v_{\ell})d^{c}(w-v_{\ell})\wedge
d\omega\wedge\eta\wedge\omega^{n-m}\right|^{2}$ $\displaystyle\leq
C_{2}\int_{\Omega}(w-v_{\ell})\eta\wedge\omega^{n-m+2},$

where $C_{2}$ is the bound for

$\int_{\Omega}(w-v_{\ell})d(w-v_{\ell})\wedge
d^{c}(w-v_{\ell})\wedge\eta\wedge\omega^{n-m+1}.$

Notice also that as $w-v_{\ell}\geq 0$,

(3.23) $-(w-v_{\ell})dd^{c}(w-v_{\ell})\wedge T\leq(w-v_{\ell})\wedge
dd^{c}v_{\ell}\wedge T.$

Combining (3.19), (3.20), (3.21), (3.22) and (3.23) we get

(3.24)
$\displaystyle\int_{\Omega}(w-v_{\ell})dd^{c}\rho\wedge\eta\wedge\omega^{n-m+1}$
$\displaystyle\leq C\left(\int_{\Omega}(w-v_{\ell})dd^{c}v_{\ell}\wedge
T\right)^{\frac{1}{2}}$
$\displaystyle\quad+C\left(\int_{\Omega}(w-v_{\ell})\eta\wedge\omega^{n-m+2}\right)^{\frac{1}{2}}$
$\displaystyle\quad+C\left(\int_{\Omega}(w-v_{\ell})\eta\wedge\omega^{n-m+2}\right)^{\frac{1}{4}}.$

Notice that by smooth sequence $\rho_{j}\downarrow\rho$ we obtain this
inequality for $\rho\in P^{*}$. To finish the proof we need to know that
passing to the limit as $\ell\to\infty$ we get the inequality in the
statement. This is known for the limits of all integrals above except for the
first integral on the right hand side. To handle this, we use the fact that
$v_{\ell}\geq v$. Then

(3.25)
$\int_{\Omega}(w-v_{\ell})dd^{c}v_{\ell}\wedge\tau\wedge\omega^{n-m}\leq\int_{\Omega}(w-v)dd^{c}v_{\ell}\wedge\tau\wedge\omega^{n-m}.$

Applying Lemma 3.14 for the right hand side, we have

(3.26)
$\displaystyle\int_{\Omega}(w-v)dd^{c}v_{\ell}\wedge\eta\wedge\omega^{n-m+1}$
$\displaystyle\leq-\int_{\Omega}d(w-v)\wedge
d^{c}v_{\ell}\wedge\eta\wedge\omega^{n-m+1}$
$\displaystyle\quad+C\left(\int_{\Omega}(w-v_{\ell})\tau\wedge\omega^{n-m+2}\right)^{\frac{1}{2}},$

Combining three inequalities (3.24), (3.25) and (3.26), we obtain

$\displaystyle\int_{\Omega}(w-v_{\ell})dd^{c}\rho\wedge\tau\wedge\omega^{n-m}$
$\displaystyle\leq C\left(-\int_{\Omega}d(w-v)\wedge
d^{c}v_{\ell}\wedge\tau\wedge\omega^{n-m}\right)^{\frac{1}{2}}$
$\displaystyle+\quad
C\left(\int_{\Omega}(w-v_{\ell})\tau\wedge\omega^{n-m+1}\right)^{\frac{1}{2}}$
$\displaystyle+\quad
C\left(\int_{\Omega}(w-v_{\ell})\tau\wedge\omega^{n-m+1}\right)^{\frac{1}{4}}.$

Letting $\ell\to\infty$ and invoking Proposition 3.11 gives

$\displaystyle\int_{\Omega}(w-v)dd^{c}\rho\wedge\eta\wedge\omega^{n-m+1}$
$\displaystyle\leq C\left(-\int_{\Omega}d(w-v)\wedge
d^{c}v\wedge\eta\wedge\omega^{n-m+1}\right)^{\frac{1}{2}}$
$\displaystyle+\quad
C\left(\int_{\Omega}(w-v)\eta\wedge\omega^{n-m+2}\right)^{\frac{1}{2}}$
$\displaystyle+\quad
C\left(\int_{\Omega}(w-v)\eta\wedge\omega^{n-m+2}\right)^{\frac{1}{4}}.$

Now the desired estimate follows from Lemma 3.13. ∎

######  Remark 3.16.

In Lemmas 3.13, 3.14, 3.15 we only need to assume that $w$ is continuous after
using one more approximation argument. In fact, it is enough to assume $w\in
P^{*}$.

For further applications we need the following lemma.

######  Lemma 3.17.

Let $u$ be a bounded $m-\omega$-sh. Suppose $\\{u_{j}\\}_{j\geq 1}$ is a
decreasing sequence of smooth $m-\omega$-sh functions with
$\|u_{j}\|_{L^{\infty}}\leq 1$ such that $u_{j}\downarrow u$ point-wise.
Assume also that all $u_{j}=u$ on a neighborhood of $\partial\Omega$. Let
$-1\leq\rho\leq 0$ be $m-\omega$-sh functions. Then, for every $0\leq s\leq
m-1$,

$\lim_{j\to+\infty}\sup_{\rho}\left\\{\int_{\Omega}(u_{j}-u)(dd^{c}\rho)^{s}\wedge\omega^{n-s}\right\\}=0.$

###### Proof.

First we can modify $\rho$ outside $K$ by $A\psi$ without changing the
integral value. Hence, we may assume that $\rho\in P^{*}$. Then, applying
repeatedly Lemma 3.15 to replace $dd^{c}\rho$ by $dd^{c}u$ we get

$\int_{\Omega}(u_{j}-u)(dd^{c}\rho)^{m-1}\wedge\omega^{n-m+1}\leq
C\sum_{s=1}^{m-1}[e_{(0,0,s)}]^{\frac{1}{2^{s}}},$

where

$e_{(0,0,s)}=\int_{\Omega}(u_{j}-u)(dd^{c}u)^{s}\wedge\omega^{n-s}.$

Thus, the proof of the lemma follows by the dominated convergence theorem
applied to each term on the right hand side. ∎

We get immediate consequences by the classical arguments. Let us define the
$(m-1)$-capacity in the class of $m-\omega$-sh functions. For a Borel set
$E\subset\Omega$,

$cap_{m-1}(E)=\sup\left\\{\int_{E}H_{m-1}(\rho):-1\leq\rho\leq 0,\;\rho\text{
is }m-\omega\text{-sh in }\Omega\right\\}.$

######  Corollary 3.18.

If $\\{u_{j}\\}_{j\geq 1}$ is a uniformly bounded decreasing sequence of
smooth $m-\omega$-sh functions such that $u_{j}\downarrow u$ ($m-\omega$-sh
function). Then, $u_{j}\to u$ in $cap_{m-1}(\bullet)$.

######  Corollary 3.19.

Every bounded $m-\omega$-sh function is quasi-continuous with respect to
$cap_{m-1}(\bullet)$.

The quasicontinuity in $(m-1)$-capacity allows us to obtain the weak
convergence for the wedge product of $m-\omega$-sh bounded functions (comp.
[BT87, Proposition 3.2]).

######  Proposition 3.20.

Let $0\leq p\leq m-1$. Let $v,u_{1},...,u_{p}$ be uniformly bounded
$m-\omega$-sh functions in $\Omega$. Let $\\{v_{\ell}\\}_{\ell\geq 1}$,
$\\{u_{s}^{j}\\}_{j\geq 1}$ be decreasing sequences of smooth $m-\omega$-sh
functions such that $v_{\ell}\downarrow v$ and $u_{s}^{j}\downarrow u_{s}$ for
each $1\leq s\leq p$. Then

$v_{j}dd^{c}u_{1}^{j}\wedge\cdots\wedge dd^{c}u_{p}^{j}\to
vdd^{c}u_{1}\wedge\cdots\wedge dd^{c}u_{p}$

as $j\to\infty$ in the sense of currents of order zero. Consequently,

$dd^{c}v_{j}\wedge dd^{c}u_{1}^{j}\wedge\cdots\wedge
dd^{c}u_{p}^{j}\wedge\omega^{n-p}\to dd^{c}v\wedge
dd^{c}u_{1}\wedge\cdots\wedge dd^{c}u_{p}\wedge\omega^{n-p}$

in the sense of currents.

###### Proof.

By the localization principle we may assume $\Omega$ is a ball and all
functions belong to $P^{*}$. Without loss of generality we may assume that
$-1\leq v_{j},u^{j}_{s},v,u\leq 0$. Denote
$S^{j}=dd^{c}u_{1}^{j}\wedge\cdots\wedge dd^{c}u_{p}^{j}$ and
$S=dd^{c}u_{1}\wedge\cdots\wedge dd^{c}u_{p}$. Both of them are currents of
order zero. Let $\chi$ be a test form with $\mbox{supp }\chi=K$. We need to
show that

$\lim_{j\to\infty}\left|\int(v_{j}S^{j}-vS)\wedge\chi\right|=0.$

We argue by induction over $p$. It is obvious for $p=0$. Assume that it is
true for $p-1\leq m-2$. This means that

$u_{1}^{j}dd^{c}u_{2}^{j}\wedge\cdots\wedge dd^{c}u^{j}_{p}\to
u_{1}dd^{c}u_{2}\wedge\cdots\wedge dd^{c}u_{p}$

in the sense of currents of order zero. Hence,

$S^{j}=dd^{c}[u_{1}^{j}dd^{c}u_{2}^{j}\wedge\cdots\wedge dd^{c}u^{j}_{p}]\to
dd^{c}[u_{1}dd^{c}u_{2}\wedge\cdots\wedge dd^{c}u_{p}]=S.$

Furthermore, by [KN16, Corollary 2.4], for a test form $\phi$, we have

(3.27) $|S^{j}\wedge\phi|\leq
C\|\phi\|_{K}\widetilde{S}^{j}\wedge\omega^{n-p},$

where $\widetilde{S}^{j}=(dd^{c}\sum_{s=1}^{p}u_{s}^{j})^{p}$ and
$\|\phi\|_{K}$ is the $C^{0}$-norm of $\phi$ on $K$. Note that
$\widetilde{S}^{j}\to\widetilde{S}=(dd^{c}\sum_{s=1}^{p}u_{s})^{p}$ as the
weak convergence holds for the wedge product of currents associated to $p$
bounded functions. Letting $j\to\infty$ one obtains

(3.28) $|S\wedge\phi|\leq C\|\phi\|_{K}\widetilde{S}\wedge\omega^{n-p}.$

We are ready to show that the conclusion holds for $p\leq m-1$. Notice that
the uniform constants $C$ below depend additionally on the uniform norm of
coefficients of $\chi$.

Let $\varepsilon>0$. By the quasi-continuity with respect to $(m-1)$-capacity
in Corollary 3.19, we can find an open set $G$ such that
$cap_{m-1}(G)<\varepsilon$ and $u_{s}^{j},u_{s},v$ are continuous on
$\Omega\setminus G$. Note that $\Omega\setminus G$ is compact in $\Omega$. It
follows from Dini’s theorem that $u^{j}_{s}\to u_{s}$ uniformly as
$j\to\infty$ on that set. We first write

(3.29) $\int(v_{j}S^{j}-vS)\wedge\chi=\int(v_{j}-v)S^{j}\wedge\chi+\int
v(S^{j}-S)\wedge\chi.$

For the first term on the right, it follows from (3.27) that

(3.30) $\left|\int(v_{j}-v)S^{j}\wedge\chi\right|\leq
C\left(\int_{\Omega\setminus
G}(v_{j}-v)\widetilde{S}^{j}\wedge\omega^{n-p}+n^{p}cap_{m-1}(G)\right),$

where we used the fact that

$\int_{G}(v_{j}-v)\widetilde{S}^{j}\wedge\omega^{n-p}\leq n^{p}cap_{m-1}(G).$

Next, we estimate the second integral in (3.29). Let $-1\leq\widetilde{v}\leq
0$ be a continuous extension of $v$ from $\Omega\setminus G$ to $\Omega$. By
induction hypothesis we know that $S^{j}\to S$ weakly, as currents. So,

(3.31) $\lim_{j\to\infty}\int\widetilde{v}(S^{j}-S)\wedge\chi=0.$

Moreover, similarly as above

$\left|\int_{G}\widetilde{v}S^{j}\wedge\chi\right|\leq Cn^{p}cap_{m-1}(G),$

and by (3.28)

$\left|\int_{G}\widetilde{v}S\wedge\chi\right|\leq
C\left|\int_{G}\widetilde{S}\wedge\omega^{n-p}\right|\leq Cn^{p}cap_{m-1}(G).$

Then,

(3.32) $\left|\int
v(S^{j}-S)\wedge\chi\right|\leq\left|\int\widetilde{v}(S^{j}-S)\wedge\chi\right|+4Cn^{p}cap_{m-1}(G).$

Combining (3.29), (3.30), (3.31) and (3.32) we conclude that

$\lim_{j\to\infty}\left|\int(v_{j}S^{j}-vS)\wedge\chi\right|\leq
C\varepsilon.$

This holds for arbitrary $\varepsilon$, so the proof of the proposition
follows. ∎

######  Remark 3.21.

The proposition showed that the wedge product of currents associated to
continuous $m-\omega$-sh functions given in [KN16, Proposition 2.11] coincides
with the one in Definition 3.4.

######  Corollary 3.22.

The wedge products

$dd^{c}u_{1}\wedge\cdots\wedge dd^{c}u_{m-1},\quad
dd^{c}u_{1}\wedge\cdots\wedge dd^{c}u_{m}\wedge\omega^{n-m}$

are symmetric with respect to bounded $m-\omega$-sh functions
$u_{1},...,u_{m-1},u_{m}$.

We can now extend the generalized Cauchy-Schwarz inequality in Lemma 2.4 and
Corollary 2.5 to the case of bounded functions. We state here a simple version
which is essential to prove the quasi-continuity later on. This is a stronger
version of Corollary 3.8. Let $-1\leq u_{1},...,u_{m-1}\leq 0$ be bounded
$m-\omega$-sh functions in $\Omega$. Denote

(3.33) $\tau=dd^{c}u_{1}\wedge\cdots\wedge dd^{c}u_{m-1}.$

######  Corollary 3.23.

Let $0\leq\phi\leq 1$ be a continuous function such that $\mbox{supp
}\phi=K\subset\subset\Omega$. Let $0\leq h\leq 1$ be a smooth $m-\omega$-sh
function. There exists a constant $C=C(\omega,K)$ such that

$\left|\int\phi dh\wedge
d^{c}\omega\wedge\tau\wedge\omega^{n-m-1}\right|^{2}\leq
C\int\phi\;\widetilde{\tau}\wedge\omega^{n-m+1},$

where $\widetilde{\tau}=\left(dd^{c}\sum_{s=1}^{m-1}u_{s}\right)^{m-1}$.

###### Proof.

Notice that the left hand side is well-defined because the coefficients of $S$
are signed Radon measures. Let $\\{u^{j}_{s}\\}_{j\geq 1}$ be a sequence of
smooth $m-\omega$-sh functions such that $u_{s}^{j}\downarrow u_{s}$ for each
$1\leq s\leq m-1$. We may assume $-1\leq u_{s}^{j}\leq 0$. Denote

$S^{j}=dd^{c}u^{j}_{1}\wedge\cdots
dd^{c}u^{j}_{m-1},\quad\widetilde{S}^{j}=(dd^{c}u_{1}^{j}+\cdots+dd^{c}u_{m-1}^{k})^{m-1}$

Applying Corollary 2.5 for $\gamma_{s}=dd^{c}u^{j}_{s}$, $s=1,...,p$, we get

$\displaystyle\left|\int\phi\;dh\wedge d^{c}\omega\wedge
S^{j}\wedge\omega^{n-m-1}\right|^{2}$ $\displaystyle\leq C\int\phi\;dh\wedge
d^{c}h\wedge(\sum_{s=1}^{m-1}\gamma_{s})^{m-1}\wedge\omega^{n-m}\times\int\phi\;\widetilde{S}^{j}\wedge\omega^{n-m+1}.$

As in the proof of Corollary 3.8, the CLN inequality implies that the first
integral inequality on the right hand side is bounded by a constant
$C=C(\omega,K)$. By Proposition 3.20 the last integral converges to
$\int\;\phi\;\widetilde{\tau}\wedge\omega^{n-m+1}.$ Therefore, the proof of
the corollary follows. ∎

##  4\. Quasi-continuity

Having the Hessian measure defined for bounded $m-\omega$-sh functions, we can
introduce the $m$-capacity (cf. [BT82]): for a Borel subset $E\subset\Omega$,

(4.1)
$cap_{m}(E)=\sup\left\\{\int_{E}(dd^{c}\rho)^{m}\wedge\omega^{n-m}:\rho\text{
is }m-\omega\text{-sh in }\Omega,-1\leq\rho\leq 0\right\\}.$

Here in fact $cap_{m}(E)=cap_{m}(E,\Omega)$ but we shall often suppress
$\Omega$ in the notation if the domain is fixed. Then, this is an inner
capacity, namely,

$cap_{m}(E)=\sup\\{cap_{m}(K):K\text{ is compact subset of }E\\}.$

######  Proposition 4.1.

Let $\Omega$ be a open set in $\mathbb{C}^{n}$ and
$cap_{m}(E)=cap_{m}(E,\Omega)$. Then,

  * (a)

If $E_{1}\subset E_{2}$, then $cap_{m}(E_{1})\leq cap_{m}(E_{2})$.

  * (b)

If $E\subset\Omega_{1}\subset\Omega_{2}$, then $cap_{m}(E,\Omega_{2})\leq
cap_{m}(E,\Omega_{1})$.

  * (c)

$cap_{m}(\cup_{j}E_{j})\leq\sum_{j}cap_{m}(E_{j})$.

  * (d)

If $E_{1}\subset E_{2}\subset\cdots$ are Borel set in $\Omega$ and
$E:=\cup_{j}E_{j}$, then $cap_{m}(E)=\lim_{j}cap_{m}(E_{j})$.

######  Definition 4.2 (Convergence in capacity).

A sequence of Borel functions $u_{j}$ in $\Omega$ is said to converge in
capacity (or in $cap_{m}(\bullet)$) to $u$ if for any $\delta>0$ and
$K\subset\subset\Omega$,

$\lim_{j\to\infty}cap_{m}(K\cap|u_{j}-u|\geq\delta)=0.$

######  Proposition 4.3.

If $\\{u_{j}\\}_{j\geq 1}$ be a uniformly bounded sequence of continuous
$m-\omega$-sh functions that decreases to a bounded $m-\omega$-sh function $u$
in $\Omega$. Then, $u_{j}$ converges to $u$ in $cap_{m}(\bullet)$.

The proof of this proposition will need the improved versions of Lemma 3.13,
3.14, 3.15 in which $\eta\wedge\omega$ is replaced by

$\tau=dd^{c}u_{1}\wedge\cdots dd^{c}u_{m-1},$

where $u_{1},...,u_{m-1}\in P^{*}$ in (3.11) and the positivity assumption
(3.12) is satisfied. This is done thanks to Corollary 3.23. Since the proofs
follow line by line those from the previous section after replacing
$\eta\wedge\omega$ by $\tau$ and applying Corollary 3.23 instead of Corollary
3.8, we only state the lemmas.

######  Lemma 4.4.

Let $\\{v_{\ell}\\}_{\ell\geq 1}$ be a sequence of smooth functions from
$P^{*}$ and let $w\in P^{*}$ be smooth. Assume $v_{\ell}\leq w$ and
$v_{\ell}\downarrow v\in P^{*}$. Then,

$\displaystyle-\int d(w-v)\wedge d^{c}v\wedge\tau\wedge\omega^{n-m}$
$\displaystyle\leq\int(w-v)dd^{c}v\wedge\tau\wedge\omega^{n-m}$
$\displaystyle+C\left(\int(w-v)\tau\wedge\omega^{n-m+1}\right)^{\frac{1}{2}},$

where the constant $C$ depends on $\omega,K,\psi$ and the uniform norm of
functions, but it is independent of $\delta$ in (3.12).

######  Lemma 4.5.

Let $v,w$ and $\\{v_{\ell}\\}$ be as in Lemma 4.4. Let $\rho$ be a bounded
$m-\omega$-sh in $\Omega$ such that $-1\leq\rho\leq 0$. Then,

$\displaystyle\int_{\Omega}(w-v)dd^{c}\rho\wedge\tau\wedge\omega^{n-m}$
$\displaystyle\leq-\int_{\Omega}d(w-v)\wedge
d^{c}\rho\wedge\tau\wedge\omega^{n-m}$
$\displaystyle\quad+C\left(\int_{\Omega}(w-v)\wedge\tau\wedge\omega^{n-m+1}\right)^{\frac{1}{2}}.$

The dependence of $C$ is the same as in the previous lemma.

Notice that as in Lemma 3.15 the strict positivity assumption (3.12) is not
needed in the following statement.

######  Lemma 4.6.

Let $v,w$ and $\\{v_{\ell}\\}$ be as in Lemma 4.4. Let $\rho$ be a bounded
$m-\omega$-sh in $\Omega$ such that $-1\leq\rho\leq 0$. Then,

$\displaystyle\int_{\Omega}(w-v)dd^{c}\rho\wedge\tau\wedge\omega^{n-m}$
$\displaystyle\leq
C\left(\int_{\Omega}(w-v)dd^{c}v\wedge\tau\wedge\omega^{n-m}\right)^{\frac{1}{2}}$
$\displaystyle\quad+C\left(\int_{\Omega}(w-v)\tau\wedge\omega^{n-m+1}\right)^{\frac{1}{2}}$
$\displaystyle\quad+C\left(\int_{\Omega}(w-v)\tau\wedge\omega^{n-m+1}\right)^{\frac{1}{4}}$

With the same dependence of $C$ as in previous lemmata.

######  Remark 4.7.

In Lemmas 4.4, 4.5, 4.6 it is enough to assume that $w\in P^{*}$ after using
one more approximation argument.

Thanks to Lemma 4.6, we can easily prove the last case $s=m$ of Lemma 3.17.

######  Lemma 4.8.

Let $u$ be a bounded $m-\omega$-sh. Suppose $\\{u_{j}\\}_{j\geq 1}$ is a
decreasing sequence of smooth $m-\omega$-sh functions with
$\|u_{j}\|_{L^{\infty}}\leq 1$ such that $u_{j}\downarrow u$ point-wise.
Assume also that all $u_{j}=u$ on a neighborhood of $\partial\Omega$. Let
$-1\leq\rho\leq 0$ be $m-\omega$-sh functions. Then,

(4.2)
$\lim_{j\to+\infty}\sup_{\rho}\left\\{\int_{\Omega}(u_{j}-u)(dd^{c}\rho)^{m}\wedge\omega^{n-m}\right\\}=0.$

###### End of proof of Proposition 4.3.

Because of (b) and (c) in Proposition 4.1 we can assume that $\Omega$ is a
ball and all functions are equal near the boundary. Let $\delta>0$. We wish to
show that

$\lim_{j\to\infty}cap_{m}(\\{u_{j}-u>\delta\\})=0.$

Argument by contradiction. Suppose that the statement were not true. Then,
there would exist $\varepsilon>0$, a subsequence
$\\{u_{j_{s}}\\}\subset\\{u_{j}\\}$ and a sequence of $m-\omega$-sh functions
$\rho_{j_{s}}$ with $-1\leq\rho_{j_{s}}\leq 0$ such that

$\limsup_{j_{s}\to+\infty}\int_{\\{u_{j_{s}}-u>\delta\\}}H_{m}(\rho_{j_{s}})\geq\varepsilon.$

On the other hand, by Markov’s inequality,

$\int_{\\{u_{j_{s}}-u>\delta\\}}H_{m}(\rho_{j_{s}})\leq\frac{1}{\delta}\int_{\Omega}(u_{j_{s}}-u)H_{m}(\rho_{j_{s}}).$

Lemma 4.8 shows that the right hand side converges to zero. This leads to a
contradiction. Thus, the proof of proposition is completed. ∎

######  Theorem 4.9.

Let $\Omega$ be a bounded open set in $\mathbb{C}^{n}$. Let $u$ be a
$m-\omega$-sh function in $\Omega$. Then, for every $\varepsilon>0$, there
exists an open subset $U\subset\Omega$ with $cap_{m}(U,\Omega)<\varepsilon$
such that $u$ restricted to $\Omega\setminus U$ is continuous.

###### Proof.

The result is local and moreover, it can be reduced to the case of bounded
functions by the property (7.5) whose proof will use only bounded functions.
We can use the classical argument in [BT82, Theorem 3.5] since there exists a
sequence of smooth $m-\omega$-sh functions that decrease to $u$ point-wise
(Proposition 2.9) and our capacity is subadditive by Proposition 4.1. ∎

We obtain the convergence in capacity for monotone sequences of uniformly
bounded functions. In particular, the smoothness assumption in Proposition 4.3
can be relaxed by yet another approximation. We recall first a classical
result in measure theory, giving a short proof for the reader’s convenience.

######  Lemma 4.10.

Let $\mu_{j}$ be a sequence of positive Radon measures with compact support in
$\Omega$. Assume that $\mu_{j}$ converges weakly to $\mu$. Let
$\Omega\supset\supset F_{1}\supset F_{2}\supset\cdots$ be a sequence of
decreasing closed subsets in $\Omega$ satisfying

$\lim_{j\to\infty}\mu(F_{j})=0.$

Then, $\lim_{j\to+\infty}\mu_{j}(F_{j})=0.$

###### Proof.

Fix $\varepsilon>0$. By the assumption there exists $j_{0}>0$ such that
$\mu(F_{j_{0}})<\varepsilon$. Using the inclusions we have
$\mu_{j}(F_{j})\leq\mu_{j}(F_{j_{0}}),$ for $j>j_{0}.$ The weak convergence
implies

$\limsup_{j\to\infty}\mu_{j}(F_{j})\leq\limsup_{j\to\infty}\mu_{j}(F_{j_{0}})\leq\mu(F_{j_{0}})<\varepsilon.$

It follows that

$0\leq\liminf_{j\to\infty}\mu_{j}(F_{j})\leq\limsup_{j\to\infty}\mu_{j}(F_{j})\leq\varepsilon.$

This holds for every $\varepsilon>0$. Thus, the conclusion follows. ∎

######  Corollary 4.11.

Let $\Omega$ be a bounded open set in $\mathbb{C}^{n}$. Let
$\\{u_{j}\\}_{j\geq 1}$ be a uniformly bounded and monotone sequence of
$m-\omega$-sh functions that either $u_{j}\downarrow u$ point-wise or
$u_{j}\uparrow u$ almost everywhere for a bounded $m-\omega$-sh function $u$
in $\Omega$. Then, $u_{j}$ converges to $u$ in $m-$capacity.

###### Proof.

The localization principle applies in both cases, we can assume that $\Omega$
is a ball and $u_{j}$’s are equal to a fixed smooth psh function in a
neighborhood of the boundary. Hence, $u-u_{j}=0$ on $\Omega\setminus K$ for a
fixed compact subset $K$. Let $\delta>0$, we wish to show that

$\lim_{j\to\infty}cap_{m}(|u-u_{j}|\geq\delta)=0.$

Arguing by contradiction, suppose that this were not true. Then, there would
exist $\varepsilon>0$ and a sequence of $m-\omega$-sh functions $\rho_{j}$
with $-1\leq\rho_{j}\leq 0$ such that

$\limsup_{j\to\infty}\int_{\\{|u-u_{j}|\geq\delta\\}}H_{m}(\rho_{j})\geq\varepsilon.$

Fix a cut-off function $\chi$ with compact support in $\Omega$ and equal $1$
on $K$. By the CLN inequality for bounded functions (Proposition 3.7) we may
assume that $\mu_{j}:=\chi H_{m}(\rho_{j})$ converges weakly to a positive
Radon measure $\mu$.

We use the quasi-continuity. Find an open set $G\subset\Omega$ such that
$cap_{m}(G)$ $\leq\varepsilon/2$ and the restrictions of $u_{j},u$ to
$\Omega\setminus G$ are continuous functions. Since $u_{j}$ is a monotone
sequence, it follows from Dini’s lemma that it converges uniformly on
$\Omega\setminus G$. In particular, the sets
$F_{j}:=\\{|u-u_{j}|\geq\delta\\}\setminus G$ are closed in $\Omega$ and
satisfy $\lim_{j\to\infty}\mu(F_{j})=0$. Hence,

$\int_{\\{|u-u_{j}|\geq\delta\\}}H_{m}(\rho_{j})\leq\int_{F_{j}}\chi
H_{m}(\rho_{j})+cap_{m}(G)\leq\mu_{j}(F_{j})+\varepsilon/2.$

Letting $j\to\infty$ and using Lemma 4.10 this leads to a contradiction. ∎

##  5\. Weak convergence

###  5.1. Convergence theorems for decreasing sequences

Let $\Omega$ be a bounded open set in $\mathbb{C}^{n}$. We have continuity of
wedge products of $m-\omega$-sh functions under decreasing sequences of
bounded functions.

######  Lemma 5.1.

Let $v,u_{1},...,u_{p}$, $1\leq p\leq m$, be a bounded $m-\omega$-sh functions
in $\Omega$. Let $\\{v^{j}\\}_{j\geq 1}$ and $\\{u^{j}_{s}\\}_{j\geq 1}$ be
uniformly bounded sequences of $m-\omega$-sh such that $v^{j}\downarrow v$ and
$u^{j}_{s}\downarrow u_{s}$ as $j\to+\infty$, for each $s=1,...,p$. Then,

  * (a)

$\lim_{j\to\infty}\mathscr{L}(u^{j}_{1},...,u^{j}_{p})=\mathscr{L}(u_{1},...,u_{p})$;

  * (b)

$\lim_{j\to+\infty}v^{j}\mathscr{L}(u_{1}^{j},...,u_{p}^{j})=v\mathscr{L}(u_{1},...,u_{p});$

where the convergence is understood in the sense of currents of order zero.

###### Proof.

(a) By the localization principle we may assume that all functions are defined
in a ball $\Omega$ and they are equal to a fixed smooth psh function $\psi$
outside $\Omega^{\prime}\subset\subset\Omega$. Let
$\\{u_{s}^{j,\delta}\\}_{\delta>0}$ be decreasing sequences of smooth
$m-\omega$-sh functions such that $u_{s}^{j,\delta}\downarrow u_{s}^{j}$ as
$\delta\to 0$. Similarly, let $\\{u_{s}^{\delta}\\}_{\delta>0}$ be
approximating sequences for $u_{s}$. We may assume that all involved functions
are negative and of uniform norm less than one.

Let $\chi$ be a test form whose $\mbox{supp }\chi=K\subset\subset\Omega$. We
consider the difference

$M_{(j,\delta)}=\int\chi\left[\mathscr{L}(u_{1}^{j,\delta},...,u_{p}^{j,\delta})-\mathscr{L}(u_{1}^{\delta},...,u_{p}^{\delta})\right].$

By Proposition 3.20 $\lim_{\delta\to
0}\mathscr{L}(u_{1}^{\delta},...,u_{p}^{\delta})=\mathscr{L}(u_{1},...,u_{p}).$
So, the proof is completed as soon as we show that

$\lim_{j\to\infty}\lim_{\delta\to 0}|M_{(j,\delta)}|=0.$

We claim that

(5.1) $|M_{(j,\delta)}|\leq
C\sum_{s=1}^{p}\int_{K}|u_{s}^{j,\delta}-u_{s}^{\delta}|(dd^{c}\rho^{j,\delta})^{p-1}\wedge\omega^{n-p+1},$

where
$\rho^{j,\delta}=\frac{1}{2p}\sum_{s=1}^{p}(u^{j,\delta}_{s}+u_{s}^{\delta})$.
In fact,

$[\mathscr{L}(u_{1}^{j,\delta},...,u_{p}^{j,\delta})-\mathscr{L}(u_{1}^{\delta},...,u_{p}^{\delta})]=\sum_{s=1}^{p}dd^{c}(u^{j,\delta}_{s}-u^{\delta}_{s})\wedge
T_{s}\wedge\omega^{n-m},$

where

$T_{s}(j,\delta)=dd^{c}u_{1}^{j,\delta}\wedge\cdots\wedge
dd^{c}u_{s-1}^{j,\delta}\wedge dd^{c}u_{s+1}^{\delta}\wedge\cdots\wedge
dd^{c}u_{p}^{\delta}.$

(Here one should use the obvious modifications for $s=1$ and $s=p$.) Notice
that $T_{s}$ is a smooth closed $(p-1,p-1)$-form. By integration by parts,

$\int\chi\omega^{n-m}\wedge dd^{c}(u_{s}^{j,\delta}-u_{s}^{\delta})\wedge
T_{p}=\int(u_{s}^{j,\delta}-u_{s}^{\delta})dd^{c}(\chi\omega^{n-m})\wedge
T_{s}.$

Since $0\leq p-1\leq m-1$, it follows from [KN16, Corollary 2.4] that

$|dd^{c}(\chi\omega^{n-m})\wedge T_{s}|\leq
2^{m}C[dd^{c}(u_{1}^{j,\delta}+\cdots+u_{p}^{\delta})]^{p-1}\wedge\omega^{n-p+1}$

for a uniform constant $C$ depending only on $\omega$ and $\chi$. This proved
the claim (5.1).

At this point we no longer have the continuity of $u_{s}^{j}$ and $u_{s}$,
however, we can make use of the quasi-continuity. Let $\varepsilon>0$. Find an
open set $G$ such that all $u^{j,\delta}_{s}$, $u^{j}_{s}$ and also
$u^{j}_{s},u_{s}$ are continuous on $\Omega\setminus G$ and
$cap_{m}(G)<\varepsilon$. We know that $\Omega\setminus G$ is compact in
$\Omega$ and by Dini’s theorem for $s=1,...,p$ we have $u_{s}^{j,\delta}\to
u_{s}^{j}$ and $u^{\delta}_{s}\to u_{s}$ uniformly as $\delta\to 0$ on that
set. Therefore,

$\displaystyle\int_{K}|u_{s}^{j,\delta}-u_{s}^{\delta}|(dd^{c}\rho^{j,\delta})^{p-1}\wedge\omega^{n-p+1}$
$\displaystyle\leq\int_{\Omega\setminus
G}|u_{s}^{j,\delta}-u_{s}^{\delta}|(dd^{c}\rho^{j,\delta})^{p-1}\wedge\omega^{n-p+1}$
$\displaystyle\quad+cap_{m}(G).$

To estimate the integral on the right hand side we use

$\displaystyle\int_{\Omega\setminus
G}|u_{s}^{j,\delta}-u_{s}^{\delta}|(dd^{c}\rho^{j,\delta})^{p-1}\wedge\omega^{n-p+1}$
$\displaystyle\leq\int_{\Omega\setminus
G}(u_{s}^{j,\delta}-u^{j}_{s})(dd^{c}\rho^{j,\delta})^{p-1}\wedge\omega^{n-p+1}$
$\displaystyle\quad+\int_{\Omega\setminus
G}(u_{s}^{\delta}-u_{s})(dd^{c}\rho^{j,\delta})^{p-1}\wedge\omega^{n-p+1}$
$\displaystyle\quad+\int_{\Omega\setminus
G}(u_{s}^{j}-u_{s})(dd^{c}\rho^{j,\delta})^{p-1}\wedge\omega^{n-p+1}.$

By the uniform convergence and then the CLN inequality (Proposition 3.7) the
first two terms go to zero as $\delta\to 0$. Moreover, $\rho^{j,\delta}$ is a
sequence of smooth $m-\omega$-sh functions decreasing to
$\rho^{j}=\frac{1}{2p}\sum_{s=1}^{p}(u^{j}_{s}+u_{s})$ as $\delta\to 0$. This
implies by Proposition 3.20 that $\mathscr{L}_{p-1}(\rho^{j,\delta})$
converges weakly to $\mathscr{L}_{p-1}(\rho^{j})$ as $\delta\to 0$. Combining
this with the continuity on $\Omega\setminus G$ we get

$\lim_{\delta\to 0}\int_{\Omega\setminus
G}(u^{j}_{s}-u_{s})\mathscr{L}_{p-1}(\rho^{j,\delta})=\int_{\Omega\setminus
G}(u^{j}_{s}-u_{s})\mathscr{L}_{p-1}(\rho^{j}).$

It follows that

$\lim_{\delta\to 0}|M_{(j,\delta)}|\leq\int_{\Omega\setminus
G}(u^{j}_{s}-u_{s})\mathscr{L}_{p-1}(\rho^{j})+\varepsilon.$

Letting $j\to+\infty$ we have by the uniform convergence,

$\lim_{j\to\infty}\lim_{\delta\to 0}|M_{(j,\delta)}|\leq\varepsilon.$

Since $\varepsilon>0$ is arbitrary, the proof is completed.

(b) For simplicity we assume that $u_{1}=\cdots u_{p}=u$ and also
$\\{u_{s}^{j}\\}=\\{u^{j}\\}$. The general case follows in the same way. Then
we write

$\mathscr{L}_{p}(u)=\mathscr{L}_{p}(u,...,u).$

Since $v^{j}$ decreases to $v$ and $\mathscr{L}_{p}(u^{j})$ converges weakly
to $\mathscr{L}_{p}(u)$ thanks to (a), any weak limit $\Theta$ of the sequence
$v^{j}\mathscr{L}_{p}(u^{j})$ satisfies

$\Theta\leq v\mathscr{L}_{p}(u).$

Hence, $v\mathscr{L}_{p}(u)-\Theta$ is a positive current. In particular,

$v\mathscr{L}_{p}(u)\wedge\omega^{m-p}-\Theta\wedge\omega^{m-p}=vH_{p}(u)-\Theta\wedge\omega^{m-p}$

is a positive Radon measure. To show the converse let $\chi\geq 0$ be a test
function in $\Omega$, we will show that

$\int\chi\Theta\wedge\omega^{m-p}\geq\int\chi vH_{p}(u).$

In fact, since $v^{j}H_{p}(u^{j})$ converges weakly to
$\Theta\wedge\omega^{m-p}$, it is enough to show

(5.2) $\lim_{j\to+\infty}\int\chi v^{j}H_{p}(u^{j})\geq\int\chi vH_{p}(u).$

Let $\varepsilon>0$ and choose an open set $G\subset\Omega$ such that
$cap_{m}(G,\Omega)\leq\varepsilon$ and $v,v_{s}$ are all continuous on
$F=\Omega\setminus G$. Since $v$ is continuous on $F$, there is a continuous
extension $g$ to $\Omega$ such that $v=g$ on $F$ with the same uniform norm.
Without loss of generality we also assume that $0\leq v,v_{s},g\leq 1$. Hence,

$\displaystyle\int\chi vH_{p}(u)$ $\displaystyle\leq\int_{F}\chi
vH_{p}(u)+cap_{m}(G)$ $\displaystyle=\int_{F}\chi gH_{p}(u)+cap_{m}(G)$
$\displaystyle\leq\int\chi gH_{p}(u)+cap_{m}(G)$
$\displaystyle=\lim_{j\to\infty}\int\chi gH_{p}(u^{j})+cap_{m}(G)$
$\displaystyle\leq\lim_{j\to\infty}\int_{F}\chi gH_{p}(u^{j})+2cap_{m}(G).$

By Dini’s theorem $v^{j}$ converges to $v=g$ uniformly on $F$, so the last
integral does not exceed

$\lim_{j\to\infty}\int_{F}\chi
v^{j}H_{p}(u^{j})+\varepsilon\leq\lim_{j\to\infty}\int\chi
v^{j}H_{p}(u^{j})+\varepsilon.$

Therefore, we have proved that

$\int\chi vH_{p}(u)\leq\lim_{j\to\infty}\int\chi
v^{j}H_{p}(u^{j})+2cap_{m}(G)+\varepsilon.$

Since $cap_{m}(G)\leq\varepsilon$ and $\varepsilon>0$ is arbitrary, the proof
of the inequality (5.2) follows, so does the one of the lemma. ∎

######  Corollary 5.2.

Let $u,v$ be bounded $m-\omega$-sh functions and $T=dd^{c}v_{1}\wedge\cdots
dd^{c}v_{m-p}\wedge\omega^{n-m}$ for bounded $m-\omega$-sh functions
$v_{1},...,v_{m-p}$, where $1\leq p\leq m$. Then,

${\bf 1}_{\\{u<v\\}}(dd^{c}\max\\{u,v\\})^{p}\wedge T={\bf
1}_{\\{u<v\\}}(dd^{c}v)^{p}\wedge T.$

Consequently,

$(dd^{c}\max\\{u,v\\})^{p}\wedge T\geq{\bf 1}_{\\{u\geq
v\\}}(dd^{c}u)^{p}\wedge T+{\bf 1}_{\\{u<v\\}}(dd^{c}v)^{p}\wedge T.$

###### Proof.

Given the weak convergence results under decreasing sequences in the above
lemma, the proof of [GZ17, Theorem 3.27] can be easily adapted to the current
case. ∎

We conclude this section by going back to the extensions of results in Section
2.4, noted in Remark 2.17. We give here only the most important statement that
will be used later.

######  Corollary 5.3.

Let $\Omega\subset\subset\mathbb{C}^{n}$ be strictly $m$-pseudoconvex. Let
$-1\leq v\leq w\leq 0$ be bounded $m-\omega$-sh functions such that
$\lim_{z\to\partial\Omega}(w-v)=0$. Let $\rho$ be a bounded $m-\omega$-sh
function such that $-1\leq\rho\leq 0$. There is a constant $C=C(\omega,n,m)$
such that

$\int_{\Omega}(w-v)^{3m}(dd^{c}\rho)^{m}\wedge\omega^{n-m}\leq
C\sum_{s=0}^{m}\int_{\Omega}(w-v)(dd^{c}v)^{s}\wedge\omega^{n-s}.$

###### Proof.

Let us replace $w$ by $w_{\varepsilon}=\max\\{w-\varepsilon,v\\}$ for
$\varepsilon>0$, so that $w_{\varepsilon}=v$ in a neighborhood of
$\partial\Omega$. If we could prove the inequality for $w_{\varepsilon}$ and
$v$, then by letting $\varepsilon\to 0$, the domination convergence theorem
would imply the required inequality. Let $\Omega^{\prime}\subset\subset\Omega$
be a smooth subdomain such that $w=v$ on $\Omega\setminus\Omega^{\prime}$.
Then the integrals on both sides will not change if we modify $v,w$ outside
$\Omega^{\prime}$. Hence, we may further assume that $w=v=\psi$ on
$\Omega\setminus\Omega^{\prime}$ with $\psi$ a smooth $m-\omega$-sh defining
function for $\Omega$.

Using the quasi-continuity it is easy to see from Lemma 5.1-(b) that for
smooth decreasing sequences $w_{j}\downarrow w$, $v_{j}\downarrow v$ and
$\rho_{j}\downarrow\rho$ we have

$\lim_{j\to\infty}\int_{\Omega}(w_{j}-v_{j})^{3m}H_{m}(\rho_{j})=\int_{\Omega}(w-v)^{3m}H_{m}(\rho),$

and for $0\leq s\leq m$,

$\lim_{j\to\infty}\int_{\Omega}(w_{j}-v_{j})H_{s}(v_{j})=\int_{\Omega}(w-v)H_{s}(v).$

Therefore, it is enough to prove the inequality for smooth functions
$v_{j}\leq w_{j}\leq 0$ and $-1\leq\rho\leq 0$. Notice that $w_{j}\to w$ and
$v_{j}\to v$ uniformly on $\Omega\setminus\Omega^{\prime}$ (this is the reason
why we modify $w,v$ near the boundary). Thus, we can follow the argument in
Remark 2.16 and conclude that the extra terms will vanish after passing to the
limit as $j\to+\infty$. Hence, the proof for the bounded functions case
follows. ∎

###  5.2. Convergence theorems for increasing sequences

With a similar proof as that of Lemma 5.1 we get

######  Lemma 5.4.

Let $1\leq p\leq m$ and $v,u_{1},...,u_{p}$ be bounded $m-\omega$-sh
functions. Suppose that $\\{v^{j}\\}_{j\geq 1}$, $\\{u_{s}^{j}\\}_{j\geq 1}$
are uniformly bounded increasing sequences of $m-\omega$-sh functions such
that $v^{j}\uparrow v$ and $u^{j}_{s}\uparrow u_{s}$ (almost everywhere) as
$j\to\infty$ for $s=1,...,p$. Then,

(5.3)
$\lim_{j\to+\infty}v^{j}\mathscr{L}(u_{1}^{j},...u_{p}^{j})=v\mathscr{L}(u_{1},...,u_{p})$

in the sense of currents of order zero.

######  Corollary 5.5.

Let $\Omega$ be a bounded open set. Let $\mathcal{U}_{m}$ be a uniformly
bounded family of $m-\omega$-sh functions in $\Omega$. Denote
$v(x)=\sup\\{v_{\alpha}(x):v_{\alpha}\in\mathcal{U}_{m}\\}$. Then, the set

$N:=\\{v<v^{*}\\}$

has zero measure with respect to any measure
$\mathscr{L}(u_{1},...,u_{m})=dd^{c}u_{1}\wedge\cdots\wedge
dd^{c}u_{m}\wedge\omega^{n-m}$, where $u_{i}$’s are bounded $m-\omega$-sh
functions. In particular, $cap_{m}(N,\Omega)=0$.

###### Proof.

By Choquet’s lemma we can reduce the argument to the case when
$\mathcal{U}_{m}$ is an increasing sequence $\\{v_{j}\\}_{j\geq 1}$ with
$w=\sup_{j}v_{j}$ and $N=\\{w<v^{*}\\}$. It follows from the proof of [GN18,
Corollary 9.9] that $w=v^{*}$ almost everywhere. Therefore, Lemma 5.4 implies
$v_{j}\mathscr{L}(u_{1},...,u_{m})$ converges weakly to
$v^{*}\mathscr{L}(u_{1},...,u_{m})$. Then, the positive currents
$(v^{*}-v_{j})\mathscr{L}(u_{1},...,u_{m})$ converge weakly to zero and hence,
for any compact set $K\subset\Omega$,

$\lim_{j\to\infty}\int_{K}(v^{*}-v_{j})\mathscr{L}(u_{1},...,u_{m})=0.$

By monotone convergence theorem,
$\lim_{j\to\infty}\int_{K}(w-v_{j})\mathscr{L}(u_{1},...,u_{m})=0.$ Therefore,
$\int_{K}(v^{*}-w)\mathscr{L}(u_{1},...,u_{m})=0$. In other words, $v^{*}=w$
a.e on $K$ with respect to $\mathscr{L}(u_{1},...,u_{m})$. The last conclusion
follows from the inner regularity of capacity. ∎

##  6\. Comparison principle

Let $\Omega$ be a bounded open set which is relatively compact in a strictly
$m$-pseudoconvex bounded domain $D$ in $\mathbb{C}^{n}$. Fix a constant ${\bf
B}$ such that on $\overline{\Omega}$,

$-{\bf B\;}\omega^{2}\leq dd^{c}\omega\leq{\bf B\;}\omega^{2},\quad-{\bf
B\;}\omega^{3}\leq d\omega\wedge d^{c}\omega\leq{\bf B\;}\omega^{3}.$

Let $\rho$ be a strictly psh function sasifying $\rho\leq 0$ and
$dd^{c}\rho\geq\omega$ in $D$. In this section we assume all function are
defined in $D$ which means that they can be approximated by a decreasing
sequence of smooth $m-\omega$-sh functions in a neighborhood of
$\overline{\Omega}$.

######  Theorem 6.1.

Let $u,v$ be bounded $m-\omega$-sh functions in $\Omega$ such that
$d=\sup_{\Omega}(v-u)>0,$ and $\liminf_{z\to\partial\Omega}(u-v)(z)\geq 0$.
Fix $0<\varepsilon<\min\\{\frac{1}{2},\frac{d}{2\|\rho\|_{\infty}}\\}$. Let us
denote for $0<s<\varepsilon_{0}:=\varepsilon^{3}/16{\bf B\;},$

$U(\varepsilon,s):=\\{u<(v+\varepsilon\rho)+S(\varepsilon)+s\\},\quad\text{where
}S(\varepsilon)=\inf_{\Omega}[u-(v+\varepsilon\rho)].$

Then,

$\int_{U(\varepsilon,s)}H_{m}(v+\varepsilon\rho)\leq\left(1+\frac{Cs}{\varepsilon^{m}}\right)\int_{U(\varepsilon,s)}H_{m}(u),$

where $C$ is a uniform constant depending on $m,n,\omega$.

###### Proof.

If $u,v$ are smooth, then the proof follows from [GN18, Lemmas 3.8, 3.9 and
3.10]. To pass from the smooth case to the bounded case we use the quasi-
continuity of $m-\omega$-sh functions and the argument as the one in [BT82,
Theorem 4.1] (see also [Ko05, Theorem 1.16]). The proof is readily adaptable
with obvious changes of notation. Here we only indicate the points of
difference that we need to take care of. Firstly, replacing $u$ by $u+\delta$
with $\delta>0$ and then letting $\delta\downarrow 0$ we may assume that
$\\{u<v\\}\subset\subset\Omega^{\prime}\subset\subset\Omega$ and $u\geq
v+\delta$ on $\Omega\setminus\Omega^{\prime}$. By restricting $u,v$ to a
smaller domain we may assume that $u,v$ are defined in a neighborhood of
$\overline{\Omega}$.

Let $\\{u_{k}\\}_{k\geq 1},\\{v_{j}\\}_{j\geq 1}$ be sequences of smooth
$m-\omega$-sh functions in a neighborhood of $\overline{\Omega}$ (Proposition
2.9) such that $u_{k}\downarrow u$ and $v_{j}\downarrow v$ point-wise in
$\overline{\Omega}$. Denote $d_{jk}=\sup_{\overline{\Omega}}(v_{j}-u_{k})$.
Then, for $j\geq k>0$ large we have

$d_{jk}\geq d/2>0.$

In fact, for small $\epsilon>0$ there exits $x\in\Omega$ such that
$d-\epsilon\leq v(x)-u(x)$. So, for $k>k_{0}$ large enough,

$d-2\epsilon\leq v(x)-u_{k}(x)\leq v_{j}-u_{k}\leq d_{jk}.$

We get the desired inequality by letting $j\to\infty$ and then $\epsilon\to
0$. Next, since $u\geq v+\delta$ on a compact set
$K=\overline{\Omega}\setminus\Omega^{\prime}$, we have $u_{k}\geq v+\delta$
for every $k\geq 1$. Since $u_{k}$ is continuous, by Hartogs’ lemma for
$\omega$-sh functions [GN18, Lemma 9.14], there is $j_{k}\geq k>0$ large
enough such that for $j\geq j_{k}$,

$v_{j}+\delta\leq u_{k}\quad\text{on }K.$

Thus, there exist subsequences of $\\{u_{k}\\}$ and $\\{v_{j}\\}$, which can
be used in the argument from [BT82, Theorem 4.1]. ∎

######  Corollary 6.2.

Let $u,v$ be bounded $m-\omega$-sh functions in a neighborhood of
$\overline{\Omega}$ such that $\liminf_{z\to\partial\Omega}(u-v)(z)\geq 0$.
Assume that $H_{m}(v)\geq H_{m}(u)$ in $\Omega$. Then, $u\geq v$ on $\Omega$.

###### Proof.

Arguing by contradiction, suppose that $\sup_{\Omega}(v-u)=d>0$. Hence, there
exist $\delta,a>0$ so small that $\sup_{\Omega}[(1+a)v-(u+\delta)]>d/2$ and
$\liminf_{z\to\partial\Omega}[(u+\delta)-(1+a)v](z)\geq 0$. Applying Theorem
6.1 for $\widetilde{u}=u+\delta$ and $\widetilde{v}=(1+a)v$, we have for
$0<s<\varepsilon_{0}$,

$\int_{U(\varepsilon,s)}H_{m}(\widetilde{v}+\varepsilon\rho)\leq\left(1+\frac{Cs}{\varepsilon^{m}}\right)\int_{U(\varepsilon,s)}H_{m}(u).$

Observe that

$H_{m}(\widetilde{v}+\varepsilon\rho)\geq(1+a)^{m}H_{m}(v)+\varepsilon^{m}H_{m}(\rho)\geq(1+a)^{m}H_{m}(u)+\varepsilon^{m}H_{m}(\rho).$

Hence, we derive from the above inequality that

$\varepsilon^{m}\int_{U(\varepsilon,s)}H_{m}(\rho)\leq 0$

for $s>0$ so small that $(1+a)^{m}\geq 1+Cs/\varepsilon^{m}$. Therefore, the
Lebesgue measure of $U(\varepsilon,s)$ is zero. This is impossible as it is
non-empty quasi-open set for $0<s<\varepsilon_{0}$. ∎

The above argument also gives

######  Corollary 6.3 (domination principle).

Let $u,v$ be bounded $m-\omega$-sh such that
$\limsup_{z\to\partial\Omega}|u(z)-v(z)|=0$ and $\int_{\\{u<v\\}}H_{m}(u)=0$.
Then, $u\geq v$ in $\Omega$.

##  7\. Polar sets and negligible sets

In this section we study the polar sets and negligible sets of $m-\omega$-sh
functions. We obtain here results analogous to those in pluripotential theory
from [BT82]. Let us first give the definitions.

######  Definition 7.1 ($m$-polar sets).

A set $E$ in $\mathbb{C}^{n}$ is $m$-polar if for each $z\in E$ there is an
open set $z\in U$ and a $m-\omega$-sh function $u$ in $U$ such that $E\cap
U\subset\\{u=-\infty\\}$.

Let $\\{u_{\alpha}\\}$ be a family of $m-\omega$-sh functions in $\Omega$
which is locally bounded from above. Then, the function

$u(z)=\sup_{\alpha}u_{\alpha}(z)$

need not be $m-\omega$-sh, but its upper semicontinuous regularization

$u^{*}(z)=\lim\sup_{x\to z}u(x)\geq u(z)$

is $m-\omega$-sh (see [GN18, Proposition 2.6-(c)]). A set of the form

(7.1) $N=\\{z\in\Omega:u(z)<u^{*}(z)\\}$

is called $m$-negligible.

Notice that a $n$-polar/$n$-negligible set is pluripolar/negligible. Clearly a
pluripolar set is a $m$-polar (or $m$-negligible sets) for every $1\leq m\leq
n$. More generally, a $m$-polar (resp. $m$-negligible) is a $(m-1)$-polar
(resp. $(m-1)$-negligible) set. An effective way to study these sets is by
extremal functions.

######  Definition 7.2.

Let $E$ be a subset of a bounded open set $\Omega\subset\mathbb{C}^{n}$. We
define

$u_{E}=u_{E,\Omega}=\sup\\{v(x):v\text{ is }m-\omega\text{-sh in
}\Omega,\;u\leq 0,\;u\leq-1\text{ on }E\\}$

By Choquet’s lemma $u_{E}$ is the limit of an increasing sequence of
$m-\omega$-sh functions. It follows from [GN18, Corollary 9.9] that
$u_{E}^{*}$ is $m-\omega$-sh and $u_{E}=u_{E}^{*}$ almost everywhere.
Moreover, $u_{E}^{*}\equiv 0$ if and only if there exists an increasing
sequence of $m-\omega$-sh functions $\\{v_{j}\\}_{j\geq 1}$ satisfying

(7.2) $v_{j}\leq 0,\quad v_{j}\leq-1\text{ on
}E,\quad\int_{\Omega}|v_{j}|dV_{2n}\leq 2^{-j}.$

######  Lemma 7.3.

Let $\Omega$ be bounded open set in $\mathbb{C}^{n}$. Then

  * (i)

If $E_{1}\subset E_{2}$, then $u_{E_{2}}\leq u_{E_{2}}$.

  * (ii)

If $E\subset\Omega_{1}\subset\Omega_{2}$, then $u_{E,\Omega_{2}}\leq
u_{E,\Omega_{1}}$.

  * (iii)

Let $K_{j}$ be non-increasing sequence of compact subset in $\Omega$ and
$K=\cap_{j}K_{j}$. Then, $u_{K_{j}}^{*}$ increases almost everywhere to
$u_{K}^{*}$.

  * (iv)

If $u_{E_{j}}^{*}\equiv 0$ and $E=\cup_{j=1}^{\infty}E_{j}$, then
$u_{E}^{*}\equiv 0$.

Suppose moreover that $\Omega$ is strictly $m$-pseudoconvex. Then

  * (v)

If $E\subset\subset\Omega$ , then $\lim_{z\to\partial\Omega}u_{E}^{*}=0$.

  * (vi)

For every set $E\subset\Omega$, $H_{m}(u_{E}^{*})\equiv 0\quad\text{ on
}\Omega\setminus\overline{E}.$

###### Proof.

The properties (i) and (ii) are obvious from the definition, and also
$\lim_{j}u_{K_{j}}\leq u_{K}^{*}$ in (iii). To prove the reverse inequality
let $v$ be a $m-\omega$-sh with $v\leq 0$ and $u\leq-1$ on $K$. For
$\varepsilon>0$, the open set $U_{\varepsilon}=\\{u<-1+\varepsilon\\}$
contains $K$. Hence, $K_{j}\subset U_{\varepsilon}$ for $j$ large enough. So,
$v-\varepsilon\leq u_{K_{j}}^{*}$. Taking supremum over all such functions $v$
we get $u_{K}-\varepsilon\leq u:=\lim_{j}u_{K_{j}}$. Letting $\varepsilon\to
0$ we obtain the conclusion. Notice again that the statement that $u=u^{*}$
almost everywhere follows from [GN18, Corollary 9.9].

(iv) Let $\varepsilon>0$. By (7.2) we can choose a sequence $v_{j}\leq 0$,
$v_{j}\leq-1$ on $E_{j}$ and $\int_{\Omega}|v_{j}|dV_{2n}\leq\varepsilon
2^{-j}$. Then, $v=\sum_{j}v_{j}$ is a $m-\omega$-sh function satisfying $v\leq
0$, $v\leq-1$ on $E$ and $\int_{\Omega}|v|dV_{2n}\leq\varepsilon$. Hence,
$u_{E}^{*}\equiv 0$.

(v) Let $\psi$ be a strictly $m-\omega$-sh defining function of $\Omega$.
Then, for $A>1$ large enough, $A\psi\leq u_{E}^{*}$. This finishes the proof.

(vi) Given the unique continuous solution of the Dirichlet problem for the
homogeneous Hessian equation [GN18, Theorem 3.15] in small balls, the result
follows from a classical balayage argument. ∎

The outer capacity $cap_{m}^{*}(\bullet)$ is defined as follows.

(7.3) $cap_{m}^{*}(E)=\inf\left\\{cap_{m}(U):E\subset U,\;U\subset\Omega\text{
is open}\right\\}.$

Then, we have basic properties which follow easily from the corresponding ones
of the capacity $cap_{m}$.

######  Proposition 7.4.

Let $\Omega$ be a bounded open set in $\mathbb{C}^{n}$. Then,

  * (i)

$cap_{m}^{*}(E_{1})\leq cap_{m}^{*}(E_{2})$ if $E_{1}\subset
E_{2}\subset\Omega$;

  * (ii)

$cap_{m}^{*}(E_{1},\Omega_{1})\geq cap_{m}^{*}(E,\Omega_{2})$ if
$E\subset\Omega_{1}\subset\Omega_{2}$;

  * (iii)

$cap_{m}^{*}(\cup_{j}E_{j})\leq\sum_{j}cap_{m}^{*}(E_{j})$.

######  Lemma 7.5.

Let $\Omega\subset\subset\mathbb{C}^{n}$ be a strictly $m$-pseudoconvex
domain. Let $E\subset\subset\Omega$ a Borel subset.

$\int_{\Omega}H_{m}(u_{E}^{*})\leq cap_{m}^{*}(E)\leq
C\sum_{s=0}^{m}\int_{\Omega}(-u_{E}^{*})H_{s}(u_{E}^{*}).$

###### Proof.

We prove first the left hand side inequality. Assume that $E=\overline{E}$ is
compact. The property Lemma 7.3-(vi) implies

$\int_{\Omega}H_{m}(u_{K}^{*})=\int_{K}H_{m}(u_{K}^{*})\leq cap_{m}(K)\leq
cap_{m}^{*}(K).$

Assume $E=G$ is an open subset. We can find an increasing sequence of compact
sets $K_{j}$ such that $\cup_{j}K_{j}=G$. It is easy to see that
$u_{K_{j}}^{*}$ decreases to $u_{G}=u_{G}^{*}$ on $\Omega$. Hence, by the weak
convergence theorem for decreasing sequences, $H_{m}(u_{K_{j}}^{*})\to
H_{m}(u_{G})$ weakly. This implies

$\int_{\Omega}H_{m}(u_{G})=\lim_{j\to\infty}\int_{\Omega}H_{m}(u_{K_{j}}^{*})\leq\lim_{j\to\infty}cap_{m}(K_{j})\leq
cap_{m}(G).$

Since $cap_{m}(G)=cap_{m}^{*}(G)$, the conclusion follows.

Now let $E$ be a Borel subset. By definition there exists a sequence of open
sets $\\{O_{j}\\}$ in $\Omega$ containing $E$ such that
$cap_{m}^{*}(E)=\lim_{j}cap_{m}(O_{j})$. Replacing $O_{j}$ by $\cap_{1\leq
s\leq j}O_{j}$ we may assume that $\\{O_{j}\\}_{j\geq 1}$ is decreasing.
Moreover, by Choquet’s lemma there exists an increasing sequence $\\{v_{j}\\}$
of negative $m-\omega$-sh functions in $\Omega$ such that $v_{j}=-1$ on $E$
and $\lim_{j}v_{j}=u_{E}$ almost everywhere on $\Omega$. Set
$G_{j}=O_{j}\cap\\{v_{j}<-1+1/j\\}$. Then, $E\subset G_{j}\subset O_{j}$ and

$v_{j}-1/j\leq u_{G_{j}}\leq u_{E}.$

So, $\lim_{j\to\infty}cap_{m}(G_{j})=cap_{m}^{*}(E)$ and $u_{G_{j}}$ increases
to $u_{E}$ almost everywhere on $\Omega$. Therefore, bythe weak convergence
for increasing sequences (Lemma 5.4),

$\int_{\Omega}H_{m}(u_{E}^{*})=\lim_{j\to\infty}\int_{\Omega}H_{m}(u_{G_{j}})\leq\lim_{j\to\infty}cap_{m}(G_{j})=cap_{m}^{*}(E).$

Thus, the proof of left hand side inequality is completed.

Next we prove the other one. Let $E\subset\subset\Omega$ be a Borel subset and
consider the sets $G_{j}$ defined as above. Then,
$\lim_{j}cap_{m}(G_{j})=cap_{m}^{*}(E)$. We also have for $0\leq s\leq m$,

$\lim_{j\to\infty}\int_{\Omega}(-u_{G_{j}})H_{s}(u_{G_{j}})=\int_{\Omega}(-u_{E}^{*})H_{s}(u_{E}^{*})$

by the weak convergence for increasing sequence again. Thus, it is enough to
prove the inequality for $E=G\subset\subset\Omega$ an open subset.

To this end let $-1\leq\rho\leq 0$ be a $m-\omega$-sh function in $\Omega$.
Since $G\subset\\{u_{G}=-1\\}$ and $u_{G}=u_{G}^{*}$ it follows that for
$q\geq 1$,

$\int_{G}H_{m}(\rho)\leq\int(-u_{G})^{q}H_{m}(\rho).$

Applying Corollary 5.3 for $v=0$ and $u=u_{G}$ we get

$\int_{\Omega}(-u_{G})^{3m}H_{m}(\rho)\leq
C\sum_{s=0}^{m}\int_{\Omega}(-u_{G})H_{s}(u_{G}).$

Taking supremum over all such functions $\rho$, we get the desired inequality.
∎

######  Remark 7.6.

For a compact set $K$ in a strictly $m$-pseudoconvex domain $\Omega$,
$cap(K,\Omega)=0$ if and only if $cap_{m}^{*}(K,\Omega)=0$.

######  Proposition 7.7.

In a strictly $m$-pseudoconvex domain $\Omega$ the following are equivalent:

  * (a)

$u_{E,\Omega}^{*}=0$.

  * (b)

$E\subset\\{u=-\infty\\}$ for a $m-\omega$-sh function $u<0$ in $\Omega$.

  * (c)

$cap_{m}^{*}(E,\Omega)=0$.

###### Proof.

(a) $\Rightarrow$ (b) follows from the property (7.2) by setting
$u=\sum_{j\geq 1}v_{j}$. Conversely, $E\subset\\{v=-\infty\\}$, where $v<0$
and $m-\omega$-sh, implies $u_{E}\geq v/j$ for $j=1,2...$. So $u_{E}=0$
outside $\\{v=-\infty\\}$ whose Lebesgue measure is zero. Hence, $u_{E}^{*}=0$
by [GN18, Corollary 9.7]. The implication (c) $\Rightarrow$ (a) follows from
the fact that $H_{m}(u_{E}^{*})\equiv 0$ and the domination principle
(Corollary 6.2) if $E$ is relatively compact in $\Omega$. The general case
follows from the countable subadditivity of $cap_{m}^{*}$ and the
corresponding property of $u_{E}^{*}$ above.

To prove $(b)\Rightarrow(c)$ let us fix an open subset $V\subset\subset\Omega$
and denote $\mathcal{O}_{j}=\\{u<-j\\}\cap V$. Let $\varepsilon>0$. We wish to
find an open subset $E\subset G\subset\Omega$ with $cap_{m}(G)<\varepsilon$.
Indeed, we have $0\geq u_{\mathcal{O}_{j}}\geq\max\\{u/j,-1\\}$, where
$u_{\mathcal{O}_{j}}$ is the relative extremal function. Then,
$u_{\mathcal{O}_{j}}\uparrow 0$ a.e on $\Omega$ by using
$\omega$-subharmonicity. Now the right hand side inequality in Lemma 7.5 gives

$cap_{m}(\mathcal{O}_{j})\leq C\sum_{s=0}^{m}e_{(0,0,s)},$

where
$e_{(0,0,s)}=\int_{\Omega}(-u_{\mathcal{O}_{j}})H_{s}(u_{\mathcal{O}_{j}})$.
Applying the weak convergence theorem for increasing convergence sequences we
get that $H_{s}(u_{\mathcal{O}_{j}})\to 0$ weakly in $\Omega$, $1\leq s\leq
m$. Furthermore, for $s=m$ and $s=0$,

$\lim_{j\to\infty}\int_{\Omega}(-u_{\mathcal{O}_{j}})H_{m}(u_{\mathcal{O}_{j}})=0=\lim_{j\to\infty}\int_{\Omega}(-u_{\mathcal{O}_{j}})\omega^{n}.$

Now we claim that for $1\leq s\leq m-1$,

(7.4)
$\lim_{j\to\infty}\int_{\Omega}(-u_{\mathcal{O}_{j}})H_{s}(u_{\mathcal{O}_{j}})=0.$

Assume this is true for a moment and let us finish the proof. The above facts
imply that

(7.5) $\lim_{j\to\infty}cap_{m}(\mathcal{O}_{j})=0.$

Take a sequence of open sets $V_{s}$ exhausting $\Omega$. Choose
$\mathcal{O}_{j_{s}}=\\{u<-j_{s}\\}\cap V_{s}$ such that
$cap_{m}(\mathcal{O}_{j_{s}})<\varepsilon/2^{s}$. Define $G=\cup_{s\geq
1}\mathcal{O}_{j_{s}}$ which is an open set containing $E$ and which has
capacity less than $\varepsilon$.

Finally, let us verify (7.4). Let $\rho$ be strictly $m-\omega$-sh defining
function for $\Omega$. Since $\mathcal{O}_{j}\subset V\subset\subset\Omega$,
we have $u_{\mathcal{O}_{j}}\geq u_{V}\geq A\rho$ for a constant $A>0$
depending only on $V,\Omega$ by the proof of Lemma 7.3-(v). Hence,
$u_{\mathcal{O}_{j}}$’s can be extended to a neighborhood $\widetilde{\Omega}$
of $\overline{\Omega}$ by $A\rho$ (see e.g., (8.9)). By the CLN inequality
there is a uniform constant $C=C(\Omega,\widetilde{\Omega})$ such that for
every $j\geq 1$,

$\int_{\Omega}H_{s}(u_{\mathcal{O}_{j}})\leq C.$

Then, for a fixed $\varepsilon>0$,

$\displaystyle\int_{\Omega}(-u_{\mathcal{O}_{j}})H_{s}(u_{\mathcal{O}_{j}})$
$\displaystyle\leq A\int_{\Omega}|\rho|H_{s}(u_{\mathcal{O}_{j}})$
$\displaystyle\leq
A\varepsilon\int_{\\{|\rho|<\varepsilon\\}}H_{s}(u_{\mathcal{O}_{j}})+A\int_{\\{|\rho|\geq\varepsilon\\}}H_{s}(u_{\mathcal{O}_{j}})$
$\displaystyle\leq
AC\varepsilon+A\int_{\\{|\rho|\geq\varepsilon\\}}H_{s}(u_{\mathcal{O}_{j}}).$

Since $H_{s}(u_{\mathcal{O}_{j}})\to 0$ weakly as $j\to\infty$ and
$\\{|\rho|\geq\varepsilon\\}\subset\Omega$ is compact, letting $j\to\infty$ we
get

$\lim_{j\to\infty}\int_{\Omega}(-u_{\mathcal{O}_{j}})H_{s}(u_{\mathcal{O}_{j}})\leq
AC\varepsilon.$

This holds for arbitrary $\varepsilon>0$, where $A,C$ are uniform constants
independent of $\varepsilon$. Hence, the proof of (7.4) is completed. ∎

######  Theorem 7.8.

$m$-negligible sets are $m$-polar.

###### Proof.

The result is local, so we may assume that all functions are defined on a
bounded strictly $m$-pseudoconvex domain $\Omega$. Thanks to the
characterization in Proposition 7.7 it is enough to show that a negligible set
$E$ has outer capacity zero. Let $\\{u_{j}\\}$ be the sequence in the
definition of the negligible set and put $u=\sup_{j}u_{j}$. By Choquet’s lemma
we may assume this is an increasing sequence. Let $\varepsilon>0$. By quasi-
continuity we can find an open set $G\subset\Omega$ such that
$cap_{m}(G)<\varepsilon$ and $u,u_{j}$’s are continuous on $F:=\Omega\setminus
G$.

Since $cap_{m}^{*}(\bullet)$ is countably subadditive, it is enough to show
that

$cap_{m}^{*}(E\cap K)=0$

for a fixed compact subset $K\subset\Omega$. Observe that for all rational
numbers $r<t$, the sets

$K_{rt}=K\cap F\cap\\{u\leq r<t\leq u^{*}\\}$

are compact, because $u$ is lower semi-continuous in $K\cap F$ and $u^{*}$ is
upper-semi continuous. Also, $(K\cap E)\setminus G$ is contained in the
countable union of such compact sets. Thus, by countable subadditivity, it
remains to verify $cap_{m}^{*}(K_{rt})=0$.

Since $u$ is lower semi-continuous on $K$, there exists a constant $c$ such
that $u\geq c$ on $K$. Denote $u_{c}=\max\\{u,c\\}$ and notice that
$K_{rt}\subset K^{\prime}_{rt}$, where

$K^{\prime}_{rt}=K\cap F\cap\\{u_{c}\leq r<t\leq u_{c}^{*}\\}$

are compact sets. Since $\\{u_{c}<u_{c}^{*}\\}$ has inner capacity $cap_{m}$
zero, we have $cap_{m}(K^{\prime}_{rt})=0$. This implies that
$cap_{m}^{*}(K^{\prime}_{rt})=0$ by Remark 7.6. ∎

We have the following analogue of Josefson’s theorem whose proof is the same
as the one of [Ko05, Theorem 1.23].

######  Theorem 7.9.

For any $m$-polar subset $E$ of $\mathbb{C}^{n}$, there exists a $m-\omega$-sh
function $h$ on $\mathbb{C}^{n}$ such that $E\subset\\{h=-\infty\\}$.

##  8\. Dirichlet problem in domains in $\mathbb{C}^{n}$

Let $\Omega$ be a bounded strictly $m$-pseudoconvex domain in
$\mathbb{C}^{n}$. The comparison principle in Corollary 6.2 coupled with the
proof of [GN18, Lemma 3.13] gives the following stability estimate for the
complex Hessian equation:

######  Proposition 8.1.

Let $u,v\in C^{0}(\overline{\Omega})$ be $m-\omega$-sh in $\Omega$ and satisfy

$H_{m}(u)=fdV_{2n},\quad H_{m}(v)=gdV_{2n}$

with $0\leq f,g\in L^{p}(\Omega)$ and $p>n/m$. Then

$\|u-v\|_{L^{\infty}}\leq\sup_{\partial\Omega}|u-v|+C\|f-g\|_{L^{p}(\Omega)}^{\frac{1}{m}},$

where $C=C(m,n,p,\Omega)$.

Let $\psi\in C^{\infty}(\partial\Omega)$. Given a smooth positive function
$f\in C^{\infty}(\overline{\Omega},\mathbb{R})$, there is always a smooth
$m-\omega$-sh subsolution $\underline{u}\in C^{\infty}(\overline{\Omega})$,
that is

(8.1) $H_{m}(\underline{u})\geq f(z),\quad\underline{u}=\psi\quad\text{on
}\partial\Omega.$