Some properties of maximal plurisubharmonic function

TL;DR

This paper investigates specific properties of maximal plurisubharmonic functions within bounded domains in complex n-space, enhancing understanding of their behavior and characteristics.

Contribution

It introduces new properties of maximal plurisubharmonic functions in bounded domains of complex space, expanding theoretical knowledge in pluripotential theory.

Findings

Characterization of maximal plurisubharmonic functions

Properties related to boundary behavior

Implications for complex analysis

Abstract

The purpose of this paper is to provide some properties of maximal plurisubharmonic functions in a bounded domain of \mathbb{C}^n

Equations

$$\underset{U}{\int }|u_j{|}^{-a}(d{d}^c{u}_j{)}^n\stackrel{j\to \infty }{⟶}0,\forall a>n-1.$$

$$\underset{j\to \infty }{\lim }\left(\underset{U\cap \{u_j>-t\}}{\int }(d{d}^c{u}_j{)}^n+\underset{U\cap \{u_j>-t\}}{\int }d{u}_j\wedge d^c{u}_j\wedge (d{d}^c{u}_j{)}^{n-1}\right)=0,$$

$$(d{d}^c\chi (u_j){)}^n\ge C{1}_{\{u_j>-t\}}(d{d}^c{u}_j{)}^n+C{1}_{\{u_j>-t\}}d{u}_j\wedge d^c{u}_j\wedge (d{d}^c{u}_j{)}^{n-1}.$$

$$\underset{j\to \infty }{\lim }\underset{U}{\int }(d{d}^c\chi (u_j){)}^n=0.$$

$$\underset{U}{\int }|u_j{|}^{-2n-1+2n\alpha }(d{d}^c{u}_j{)}^n\stackrel{j\to \infty }{⟶}0,\forall 0<\alpha <\frac{1}{2n}.$$

$$\underset{U}{\int }|u_j{|}^{-n+n\beta }(d{d}^c{u}_j{)}^n\le C_{\beta },\forall j>0,\forall 0<\beta <\frac{1}{n}.$$

$$\underset{U\cap \{u_j>-M\}}{\int }(d{d}^c{u}_j{)}^n\stackrel{j\to \infty }{⟶}0,$$

$$\forall k\ge k_0,\exists l_0>0:\underset{j\to \infty }{\mathrm{limsup}}\underset{U\cap \{-k-\frac{1}{l}<u_j<-k+\frac{1}{l}\}}{\int }d{u}_j\wedge d^c{u}_j\wedge (d{d}^c{u}_j{)}^{n-1}<\frac{ϵ}{l},\forall l>l_0.$$

$$(d{d}^c{\chi }_{k,l}(u){)}^n\stackrel{l\to \infty }{⟶}(d{d}^c\max \{u,-k\}{)}^n$$

$$\underset{U\cap \{u_j>-M\}}{\int }(d{d}^c{u}_j{)}^n\stackrel{j\to \infty }{⟶}0,$$

$$\underset{j\to \infty }{\mathrm{limsup}}\underset{U\cap \{-k-1<u_j<-k\}}{\int }d{u}_j\wedge d^c{u}_j\wedge (d{d}^c{u}_j{)}^{n-1}\stackrel{k\to \infty }{⟶}0.$$

$$\underset{U\cap \{u_j>-M\}}{\int }(d{d}^c{u}_j{)}^n\stackrel{j\to \infty }{⟶}0,$$

$$\underset{j\to \infty }{\mathrm{limsup}}\underset{U\cap \{u_j<-k\}}{\int }{e}^{n(u_j+k)}d{u}_j\wedge d^c{u}_j\wedge (d{d}^c{u}_j{)}^{n-1}\stackrel{k\to \infty }{⟶}0.$$

Full text

1. 1 A class of maximal plurisubharmonic functions
2. 2 Proof of the main theorems
3. 3 Relation between some class of maximal plurisubharmonic functions

Introduction

Let $\Omega\subset\mathbb{C}^{n}$ be a bounded domain $(n\geq 2)$. A function $u\in PSH(\Omega)$ is called maximal if for every open set $G\Subset\Omega$, and for each upper semicontinuous function $v$ on $\bar{G}$ such that $v\in PSH(G)$ and $v|_{\partial G}\leq u|_{\partial G}$, we have $v\leq u$. There are some equivalent descriptions of maximality which have been presented in [Sad81], [Kli91].

For the convenience, we denote

$MPSH(\Omega)=\{u\in PSH(\Omega)|u\mbox{ is maximal }\}$,

$MPSH_{loc}(\Omega)=\{u\in PSH(\Omega)|\forall z\in\Omega:u\in MPSH(U)\mbox{ for some }z\in U\Subset\Omega\}$.

The following question was given by Blocki [Blo04], [DGZ16]:

Question 0.1**.**

Is $MPSH(\Omega)$ equal to $MPSH_{loc}(\Omega)$? (or is maximality a local notion?)

Denote by $D(\Omega)$ the domain of definition of Monge-Ampère operator in $\Omega$. By [Blo04], a function $u\in D(\Omega)$ is maximal iff $(dd^{c}u)^{n}=0$. Moreover, it follows from [Blo06] that to belong to the class $D$ is a local property. Then

$MPSH(\Omega)\cap D(\Omega)=MPSH_{loc}(\Omega)\cap D(\Omega)$.

In the general case, the question 0.1 is still open. It raises another question

Question 0.2**.**

What information can be obtained from the condition $u\in MPSH_{loc}(\Omega)$?

In this paper, we study on some subclass of $MPSH(\Omega)$ and use results on it to show some properties of the class $MPSH_{loc}(\Omega)$.

Our main results are following:

Theorem 0.3**.**

If $u,v\in MPSH_{loc}(\Omega)$ and $\chi:\mathbb{R}\rightarrow\mathbb{R}$ is a convex non-decreasing function then $(z,w)\mapsto\chi(u(z)+v(w))\in MPSH(\Omega\times\Omega)$.

Theorem 0.4**.**

Let $u$ be a negative maximal plurisubharmonic function in $\Omega$ and let $U,\tilde{U}$ be open subset of $\Omega$ such that $U\Subset\tilde{U}\Subset\Omega$. Assume that $u_{j}\in PSH^{-}(\tilde{U})\cap C(\tilde{U})$ is decreasing to $u$ in $\tilde{U}$. Then

[TABLE]

Acknowledgements**.**

I am thankful to Nguyen Quang Dieu and Nguyen Xuan Hong for useful discussion and comments.

1. A class of maximal plurisubharmonic functions

We say that a function $u\in PSH^{-}(\Omega)$ has $M_{1}$ property iff for every open set $U\Subset\Omega$, there are $u_{j}\in PSH^{-}(U)\cap C(U)$ such that $u_{j}$ is decreasing to $u$ in $U$ and

[TABLE]

for any $t>0$. We denote by $M_{1}PSH(\Omega)$ the set of negative plurisubharmonic functions in $\Omega$ satisfying $M_{1}$ property.

If $\chi:\mathbb{R}\rightarrow\mathbb{R}$ is a convex non-decreasing function, we denote $MPSH_{\chi}(\Omega)$ the set of negative plurisubharmonic functions in $\Omega$ such that $\chi(u)\in MPSH(\Omega)$.

Theorem 1.1**.**

Let $\Omega$ be a bounded domain in $\mathbb{C}^{n}$ and $u\in PSH^{-}(\Omega)$. Then the following conditions are equivalent

• (i)

$u\in M_{1}PSH(\Omega)$.

• (ii)

$u\in MPSH_{\chi}(\Omega)$* for any convex non-decreasing function $\chi:\mathbb{R}\rightarrow\mathbb{R}$.*

• (iii)

For any open sets $U,\tilde{U}$ such that $U\Subset\tilde{U}\Subset\Omega$, for any $u_{j}\in PSH^{-}(\tilde{U})\cap C(\tilde{U})$ such that $u_{j}$ is decreasing to $u$ in $\tilde{U}$, we have

$\lim\limits_{j\to\infty}\left(\int\limits_{U}|u_{j}|^{-a}(dd^{c}u_{j})^{n}+\int\limits_{U}|u_{j}|^{-a-1}du_{j}\wedge d^{c}u_{j}\wedge(dd^{c}u_{j})^{n-1}\right)=0,$**

for all $a>n-1$.

In particular, $M_{1}$ property is a local notion and $M_{1}PSH(\Omega)\subset MPSH(\Omega)$.

Proof.

$(iii\Rightarrow i)$: Obvious.

$(i\Rightarrow ii)$:

Assume that $U\Subset\tilde{U}\Subset\Omega$. Let $u_{j}\in PSH^{-}(U)\cap C(U)$ such that $u_{j}$ is decreasing to $u$ in $U$ and the condition (2) is satisfied.

If $\chi$ is smooth and $\chi$ is constant in some interval $(-\infty,-m)$ then

\begin{array}[]{ll}(dd^{c}\chi(u_{j}))^{n}&=(\chi^{\prime}(u_{j}))^{n}(dd^{c}u_{j})^{n}+n\chi^{\prime\prime}(u_{j})(\chi^{\prime}(u_{j}))^{n-1}du_{j}\wedge d^{c}u_{j}\wedge(dd^{c}u_{j})^{n-1}\\[12.0pt] &\leq C\boldsymbol{1}_{\{u_{j}>-t\}}(dd^{c}u_{j})^{n}+C\boldsymbol{1}_{\{u_{j}>-t\}}du_{j}\wedge d^{c}u_{j}\wedge(dd^{c}u_{j})^{n-1},\end{array}

where $C,t>0$ depend only on $\chi$. Hence

$\int\limits_{U}(dd^{c}\chi(u_{j}))^{n}\stackrel{{\scriptstyle j\to\infty}}{{\longrightarrow}}0.$

Then, by [Sad81] (see also [Ceg09]), $\chi(u)$ is maximal on $U$ for any open set $U\Subset\Omega$. Thus $\chi(u)\in MPSH(\Omega)$.

In the general case, for any convex non-decreasing function $\chi$, we can find $\chi_{l}\searrow\chi$ such that $\chi_{l}$ is smooth, convex and $\chi|_{(-\infty,-m)}=const$ for some $m$. By above argument, $\chi_{l}\in MPSH(\Omega)$ for any $l\in\mathbb{N}$. Hence $\chi(u)\in MPSH(\Omega).$

$(ii\Rightarrow iii)$:

For any $0<\alpha<\frac{1}{n}$, the function

$\Phi_{\alpha}(t)=-(-t)^{\alpha}$

is convex and non-decreasing in $\mathbb{R}^{-}$. Assume that $u$ satisfies (ii), we have $\Phi_{\alpha}\in MPSH(\Omega)$.

By [Bed93] (see also [Blo09]), for any $0<\alpha<\frac{1}{n}$, we have $\Phi_{\alpha}(u)\in D(\Omega)$. Then, for any $u_{j}\in PSH^{-}(\tilde{U})\cap C(\tilde{U})$ such that $u_{j}$ is decreasing to $u$ in $\tilde{U}$, we have

$\int\limits_{U}(dd^{c}\Phi_{\alpha}(u_{j}))^{n}\stackrel{{\scriptstyle j\to\infty}}{{\longrightarrow}}0,\forall 0<\alpha<\frac{1}{n},$

and it implies (iii).

Finally, by using $(i\Leftrightarrow iii)$, we conclude that $M_{1}$ property is a local notion. ∎

The following proposition is an immediately corollary of Theorem 1.1

Proposition 1.2**.**

Let $\Omega$ be a bounded domain in $\mathbb{C}^{n}$.

• (i)

If $u\in M_{1}PSH(\Omega)$ then $\chi(u)\in M_{1}PSH(\Omega)$ for any convex non-decreasing function $\chi:\mathbb{R}^{-}\rightarrow\mathbb{R}^{-}$.

• (ii)

If $u_{j}\in M_{1}PSH(\Omega)$ and $u_{j}$ is decreasing to $u$ then $u\in M_{1}PSH(\Omega)$.

• (iii)

Let $u\in PSH^{-}(\Omega)\cap C^{2}(\Omega\setminus F)$, where $F=\{z:u(z)=-\infty\}$ is closed. If

$(dd^{c}u)^{n}=du\wedge d^{c}u\wedge(dd^{c}u)^{n-1}=0$**

in $\Omega\setminus F$ then $u\in M_{1}PSH(\Omega)$.

In some special cases, we can easily check $M_{1}$ property by the following criteria

Proposition 1.3**.**

Let $\Omega$ be a bounded domain in $\mathbb{C}^{n}$. Let $\chi:\mathbb{R}\rightarrow\mathbb{R}$ be a smooth convex increasing function such that $\chi^{\prime\prime}(t)>0$ for any $t\in\mathbb{R}$. Assume also that $\chi$ is lower bounded. If $u\in PSH^{-}(\Omega)$ and $\chi(u)\in MPSH(\Omega)$ then $u\in M_{1}PSH(\Omega)$.

Proof.

Let $U\Subset\tilde{U}\Subset\Omega$ and $u_{j}\in PSH(\tilde{U})\cap C(\tilde{U})$ such that $u_{j}$ is decreasing to $u$. Then

$dd^{c}(\chi(u_{j}))=\chi^{\prime}(u_{j})dd^{c}u_{j}+\chi^{\prime\prime}(u_{j})du_{j}\wedge d^{c}u_{j}$

and

$(dd^{c}\chi(u_{j}))^{n}=(\chi^{\prime}(u_{j}))^{n}(dd^{c}u_{j})^{n}+n\chi^{\prime\prime}(u_{j})(\chi^{\prime}(u_{j}))^{n-1}du_{j}\wedge d^{c}u_{j}\wedge(dd^{c}u_{j})^{n-1}.$

For any $t>0$, there exists $C>0$ depending only on $t$ and $\chi$ such that

[TABLE]

Note that $\chi(u)\in D(\Omega)\cap MPSH(\Omega)$. Hence

[TABLE]

Combining (3) and (4), we have

$\lim\limits_{j\to\infty}\left(\int\limits_{U\cap\{u_{j}>-t\}}(dd^{c}u_{j})^{n}+\int\limits_{U\cap\{u_{j}>-t\}}du_{j}\wedge d^{c}u_{j}\wedge(dd^{c}u_{j})^{n-1}\right)=0.$

Thus $u\in M_{1}PSH(\Omega)$. ∎

Example 1.4**.**

*(i) If $u$ is a negative plurisubharmonic function in $\Omega\subset\mathbb{C}^{n}$ depending only on $n-1$ variables then $u$ has $M_{1}$ property.

(ii) If $f:\Omega\rightarrow\mathbb{C}^{n}$ is a holomorphic mapping of rank $<n$ then $(dd^{c}|f|^{2})^{n}=0$ (see, for example, in [Ras98]). Then, by Proposition 1.3, $\log|f|\in M_{1}PSH(\Omega)$ if it is negative in $\Omega$.*

Question 1.5**.**

Does Proposition 1.3 still hold if the assumption “$\chi$ is lower bounded” is removed from it?

2. Proof of the main theorems

2.1. Proof of Theorem 0.3

Without loss of generality, we can assume that $u,v\in PSH^{-}(\Omega)$.

If $u,v\in MPSH_{loc}(\Omega)$ then for any $z_{0},w_{0}\in\Omega$, there are hyperconvex domains $U,\tilde{U},V,\tilde{V}$ such that $z_{0}\in U\Subset\tilde{U}\Subset\Omega$, $w_{0}\in V\Subset\tilde{V}\Subset\Omega$, $u\in MPSH(\tilde{U})$ and $v\in MPSH(\tilde{V})$. We need to show that $u(z)+v(w)$ have $M_{1}$ property in $U\times V$.

Let $u_{j}\in PSH^{-}(\tilde{U})\cap C(\tilde{U})$ and $v_{j}\in PSH^{-}(\tilde{V})\cap C(\tilde{V})$ such that $u_{j}$ is decreasing to $u$ in $\tilde{U}$ and $v_{j}$ is decreasing to $v$ in $\tilde{V}$. By [Wal68], there are $\tilde{u}_{j}\in PSH^{-}(\tilde{U})\cap C(\tilde{U})$, $\tilde{v}_{j}\in PSH^{-}(\tilde{V})\cap C(\tilde{V})$ such that

$\begin{cases}\tilde{u_{j}}=u_{j}\quad\mbox{ in }\tilde{U}\setminus U,\\ \tilde{v_{j}}=v_{j}\quad\mbox{ in }\tilde{V}\setminus V,\\ (dd^{c}\tilde{u}_{j})^{n}=0\quad\mbox{ in }U,\\ (dd^{c}\tilde{v}_{j})^{n}=0\quad\mbox{ in }V.\end{cases}$

By the maximality of $u$ and $v$, we conclude that $\tilde{u}_{j}$ is decreasing to $u$ in $\tilde{U}$ and $\tilde{v}_{j}$ is decreasing to $v$ in $\tilde{V}$. In $U\times V$, we have

$(dd^{c}(\tilde{u}_{j}(z)+\tilde{v}_{j}(w)))^{2n}=C^{n}_{2n}(dd^{c}\tilde{u}_{j})^{n}_{z}\wedge(dd^{c}\tilde{v}_{j})^{n}_{w}=0$

$d(\tilde{u}_{j}(z)+\tilde{v}_{j}(w))\wedge d^{c}(\tilde{u}_{j}(z)+\tilde{v}_{j}(w))\wedge(dd^{c}(\tilde{u}_{j}(z)+\tilde{v}_{j}(w)))^{2n-1}$

$=C^{n-1}_{2n-1}d_{z}\tilde{u}_{j}\wedge d^{c}_{z}\tilde{u}_{j}\wedge(dd^{c}\tilde{u}_{j})^{n-1}_{z}\wedge(dd^{c}\tilde{v}_{j})^{n}_{w}$ $+C^{n-1}_{2n-1}d_{w}\tilde{v}_{j}\wedge d^{c}_{w}\tilde{v}_{j}\wedge(dd^{c}\tilde{v}_{j})^{n-1}_{w}\wedge(dd^{c}\tilde{u}_{j})^{n}_{z}$

$=0.$

Then $u(z)+v(w)$ has $M_{1}$ property in $U\times V$. By Theorem 1.1, $M_{1}$ property is a local notion. Hence $u(z)+v(w)\in M_{1}PSH(\Omega\times\Omega)$. And it implies that $u(z)+v(w)\in MPSH_{\chi}(\Omega\times\Omega)$ for any convex non-decreasing function $\chi$.

2.2. Proof of Theorem 0.4

Let $v=|z_{1}|^{2}+...+|z_{n-1}|^{2}+x_{n}+y_{n}-M$, where $M=\sup\limits_{\Omega}(|z|^{2}+|x_{n}|+|y_{n}|)$. Then $v\in MPSH(\Omega)$. By Theorem 0.3, $\chi(u(z)+v(w))\in MPSH(\Omega\times\Omega)$ for any convex non-decreasing function $\chi$.

By [Bed93],[Blo09], for any $0<\alpha<\frac{1}{2n}$, we have $\Phi_{\alpha}(u(z)+v(w))\in D(\Omega\times\Omega)$, where $\Phi_{\alpha}$ is defined as in the proof of Theorem 1.1.

Then

$\int\limits_{U\times U}(dd^{c}\Phi(u_{j}(z)+v(w)))^{2n}\stackrel{{\scriptstyle j\to\infty}}{{\longrightarrow}}0,$

for any $0<\alpha<\frac{1}{2n}$. Hence

[TABLE]

Moreover, $\Phi_{\beta}(u)\in D(\Omega)$ for any $0<\beta<\frac{1}{n}$. Then, for any $0<\beta<\frac{1}{n}$, there is $C_{\beta}>0$ such that

$\int\limits_{U}(dd^{c}\Phi_{\beta}(u_{j}))^{n}\leq C_{\beta},\,\forall j>0.$

Hence

[TABLE]

Combining (5), (6) and using Hölder inequality, we obtain (1).

3. Relation between some class of maximal plurisubharmonic functions

Let $\Omega$ be a bounded domain in $\mathbb{C}^{n}$. Let $u\in PSH(\Omega)$. If there exists a sequence of convex non-decreasing functions $\chi^{m}:\mathbb{R}\rightarrow\mathbb{R}$ such that

• •

$\chi^{m}$ is lower bounded for every $m$,

• •

$\chi^{m}$ is decreasing to $Id$ as $m\rightarrow\infty$,

• •

$(dd^{c}\chi^{m}(u))^{n}\stackrel{{\scriptstyle m\to\infty}}{{\longrightarrow}}0$ in the weak sense,

then, by [Sad81], $u$ is maximal. We are interested in the following question

Question 3.1**.**

If $u$ is maximal, does there exist a sequence of convex non-decreasing function $\chi^{m}$ satisfying above conditions? If it exists, how to find it?

In this section we discuss about relation between some class of maximal plurisubharmonic functions in a bounded domain $\Omega$ in $\mathbb{C}^{n}$. It can be seen as the first step in approaching Question 3.1.

Assume that $\chi:\mathbb{R}\rightarrow\mathbb{R}$ is a smooth convex function such that $\chi|_{(-\infty,-2)}=-1$, $\chi|_{(0,\infty)}=Id_{(0,\infty)}$ and $\chi^{\prime\prime}(-1)>0$. We denote

$M_{2}PSH(\Omega)=\{u\in PSH^{-}(\Omega)|(dd^{c}\max\{u,-k\})^{n}\stackrel{{\scriptstyle weak}}{{\longrightarrow}}0\mbox{ as }k\rightarrow\infty\}$,

$M_{3}PSH(\Omega)=\{u\in PSH^{-}(\Omega)|(dd^{c}\chi_{k}(u))^{n}\stackrel{{\scriptstyle weak}}{{\longrightarrow}}0\mbox{ as }k\rightarrow\infty\}$,

$M_{4}PSH(\Omega)=\{u\in PSH^{-}(\Omega)|(dd^{c}\log(e^{u}+\dfrac{1}{k}))^{n}\stackrel{{\scriptstyle weak}}{{\longrightarrow}}0\mbox{ as }k\rightarrow\infty\},$

where $\chi_{k}(t)=\chi(t+k)-k$. The main result of this section is following

Theorem 3.2**.**

$M_{2}PSH(\Omega)\subset M_{3}PSH(\Omega)$* and $M_{3}PSH(\Omega)=M_{4}PSH(\Omega).$*

First, we introduce some characteristics of $M_{2}PSH(\Omega),M_{3}PSH(\Omega)$ and $M_{4}PSH(\Omega)$.

Proposition 3.3**.**

Let $u\in PSH^{-}(\Omega)$. Assume that $u_{j}\in PSH^{-}(\Omega)\cap C(\Omega)$ is decreasing to $u$. Then the following conditions are equivalent

• (i)

$u\in M_{2}PSH(\Omega)$.

• (ii)

If $U\Subset\Omega$ then for every $M>0$

[TABLE]

and for any $\epsilon>0$, there is $k_{0}>0$ such that

[TABLE]

Proof.

For any $k,l>0$, we denote

$\chi_{k,l}(t)=\dfrac{1}{l}\chi((t+k-\dfrac{1}{l})l)-k+\dfrac{1}{l}.$

Then $\chi_{k,l}(u)$ converges to $\max\{u,-k\}$ as $l\rightarrow\infty$. Moreover

$\chi_{k,l}(u)\geq\max\{u,-k\}+O(\dfrac{1}{l}).$

Hence, by [Ceg09],

[TABLE]

in the weak sense.

For any $j>0$, we have

$(dd^{c}\chi_{k,l}(u_{j}))^{n}=(\chi^{\prime}((u_{j}+k-\dfrac{1}{l})l))^{n}(dd^{c}u_{j})^{n}+nl\chi^{\prime\prime}((u_{j}+k-\dfrac{1}{l})l)(\chi^{\prime}((u_{j}+k-\dfrac{1}{l})l))^{n-1}du_{j}\wedge d^{c}u_{j}\wedge(dd^{c}u_{j})^{n-1}.$

Then, there are $C_{1},C_{2},\delta>0$ depending only on $\chi$ such that

$(dd^{c}\chi_{k,l}(u_{j}))^{n}\leq C_{1}\boldsymbol{1}_{\{u_{j}>-k-\frac{1}{l}\}}(dd^{c}u_{j})^{n}+C_{1}l\boldsymbol{1}_{\{-k-\frac{1}{l}<u_{j}<-k+\frac{1}{l}\}}du_{j}\wedge d^{c}u_{j}\wedge(dd^{c}u_{j})^{n-1},$

and

$(dd^{c}\chi_{k,l}(u_{j}))^{n}\geq C_{2}\boldsymbol{1}_{\{u_{j}>-k+\frac{1}{l}\}}(dd^{c}u_{j})^{n}+C_{2}l\boldsymbol{1}_{\{-k-\frac{\delta}{l}<u_{j}<-k+\frac{\delta}{l}\}}du_{j}\wedge d^{c}u_{j}\wedge(dd^{c}u_{j})^{n-1}.$

Moreover,

$(dd^{c}\chi_{k,l}(u_{j}))^{n}\stackrel{{\scriptstyle j\to\infty}}{{\longrightarrow}}(dd^{c}\chi_{k,l}(u))^{n}$

in the weak sense.

Then $(ii)$ is equivalent to

$\limsup\limits_{l\to\infty}\int\limits_{U}(dd^{c}\chi_{k,l}(u))^{n}\stackrel{{\scriptstyle k\to\infty}}{{\longrightarrow}}0,$

for any $U\Subset\Omega$.

Hence, by (9), we conclude that $(i)$ is equivalent to $(ii)$. ∎

Proposition 3.4**.**

Let $u\in PSH^{-}(\Omega)$. Assume that $u_{j}\in PSH^{-}(\Omega)\cap C(\Omega)$ is decreasing to $u$. Then the following conditions are equivalent

• (i)

$u\in M_{3}PSH(\Omega)$.

• (ii)

If $U\Subset\Omega$ then

[TABLE]

and

[TABLE]

The proof of Proposition 3.4 is similar to the proof of Proposition 3.3. We leave the details to the reader.

In [Sad12], Sadullaev has proved a characteristic of $M_{4}PSH(\Omega)$:

Theorem 3.5**.**

Let $u\in PSH^{-}(\Omega)$. Denote $v=e^{u}$. Then $u\in M_{4}PSH(\Omega)$ iff

$v(dd^{c}v)^{n}-ndv\wedge d^{c}v\wedge(dd^{c}v)^{n-1}=0,$**

and

$\lim\limits_{t\to 0}\dfrac{1}{t^{n}}\int\limits_{U\cap\{v<t\}}(dd^{c}v)^{n}=0,$**

for any $U\Subset\Omega$.

Using Theorem 3.5, it is easy to show that

Proposition 3.6**.**

Let $u\in PSH^{-}(\Omega)$. Assume that $u_{j}\in PSH^{-}(\Omega)\cap C(\Omega)$ is decreasing to $u$. Then the following conditions are equivalent

• (i)

$u\in M_{4}PSH(\Omega)$.

• (ii)

If $U\Subset\Omega$ then

[TABLE]

and

[TABLE]

Now, by using Propositions 3.3, 3.4 and 3.6, we will prove Theorem 3.2.

Proof of Theorem 3.2.

Let $u\in PSH^{-}(\Omega)$. By replacing $\Omega$ by an exhaustive sequence of relative compact subsets of $\Omega$, we can assume that there exists a sequence $u_{j}\in PSH^{-}(\Omega)\cap C(\Omega)$ such that $u_{j}$ decreasing to $u$ in $\Omega$.

Assume that $u\in M_{2}PSH(\Omega)$. By Proposition 3.3, for any $\epsilon>0$, there exist $k_{0}>0$ such that

$\forall k\geq k_{0},\exists l_{0}>0:\limsup\limits_{j\to\infty}\int\limits_{U\cap\{-k-\frac{1}{l}<u_{j}<-k+\frac{1}{l}\}}du_{j}\wedge d^{c}u_{j}\wedge(dd^{c}u_{j})^{n-1}<\dfrac{\epsilon}{l},\forall l>l_{0}.$

Let $k>k_{0}$. By Besicovitch’s covering theorem [Matt95], there are $k_{1},l_{1},...,k_{m},l_{m}...>0$ such that

$\limsup\limits_{j\to\infty}\int\limits_{U\cap\{-k_{m}-\frac{1}{l_{m}}<u_{j}<-k_{m}+\frac{1}{l_{m}}\}}du_{j}\wedge d^{c}u_{j}\wedge(dd^{c}u_{j})^{n-1}<\dfrac{\epsilon}{l_{m}},\forall m>0,$

and

$\boldsymbol{1}_{(-k-1,-k)}\leq\sum\limits_{m}\boldsymbol{1}_{(-k_{m}-\frac{1}{l_{m}},-k_{m}+\frac{1}{l_{m}})}=\sum\limits_{m}\dfrac{2}{l_{m}}\leq C,$

where $C>0$ is a universal constant. Hence

$\limsup\limits_{j\to\infty}\int\limits_{U\cap\{-k-1<u_{j}<-k\}}du_{j}\wedge d^{c}u_{j}\wedge(dd^{c}u_{j})^{n-1}\leq C\epsilon.$

By Proposition 3.4, we get $u\in M_{3}PSH(\Omega)$.

Thus $M_{2}PSH(\Omega)\subset M_{3}PSH(\Omega)$.

Now, assume that $u$ is an arbitrary element of $M_{3}PSH(\Omega)$. By Proposition 3.4, for any $\epsilon>0$, there exists $k_{0}>0$ such that

$\limsup\limits_{j\to\infty}\int\limits_{U\cap\{-k-1<u_{j}<-k\}}du_{j}\wedge d^{c}u_{j}\wedge(dd^{c}u_{j})^{n-1}<\epsilon.$

for every $k>k_{0}$. Then

$\limsup\limits_{j\to\infty}\int\limits_{U\cap\{u_{j}<-k\}}e^{n(u_{j}+k)}du_{j}\wedge d^{c}u_{j}\wedge(dd^{c}u_{j})^{n-1}$

$\leq\sum\limits_{l=0}^{\infty}\limsup\limits_{j\to\infty}\int\limits_{U\cap\{-k-l-1<u_{j}<-k-l\}}e^{n(u_{j}+k)}du_{j}\wedge d^{c}u_{j}\wedge(dd^{c}u_{j})^{n-1}$

$\leq\sum\limits_{l=0}^{\infty}\limsup\limits_{j\to\infty}\int\limits_{U\cap\{-k-l-1<u_{j}<-k-l\}}e^{-nl}du_{j}\wedge d^{c}u_{j}\wedge(dd^{c}u_{j})^{n-1}$

$\leq\epsilon\sum\limits_{l=0}^{\infty}e^{-nl},$

for any $k>k_{0}$. Hence

$\limsup\limits_{j\to\infty}\int\limits_{U\cap\{u_{j}<-k\}}e^{n(u_{j}+k)}du_{j}\wedge d^{c}u_{j}\wedge(dd^{c}u_{j})^{n-1}\stackrel{{\scriptstyle k\to\infty}}{{\longrightarrow}}0.$

By Proposition 3.6, we get $u\in M_{4}PSH(\Omega)$.

Thus $M_{3}PSH(\Omega)\subset M_{4}PSH(\Omega)$.

Conversely, if $u\in M_{4}PSH(\Omega)$ then

$\limsup\limits_{j\to\infty}\int\limits_{U\cap\{-k-1<u_{j}<-k\}}du_{j}\wedge d^{c}u_{j}\wedge(dd^{c}u_{j})^{n-1}$

$\leq e^{n}\limsup\limits_{j\to\infty}\int\limits_{U\cap\{-k-1<u_{j}<-k\}}e^{n(u_{j}+k)}du_{j}\wedge d^{c}u_{j}\wedge(dd^{c}u_{j})^{n-1}$

$\leq e^{n}\limsup\limits_{j\to\infty}\int\limits_{U\cap\{u_{j}<-k\}}e^{n(u_{j}+k)}du_{j}\wedge d^{c}u_{j}\wedge(dd^{c}u_{j})^{n-1}$

$\stackrel{{\scriptstyle k\to\infty}}{{\longrightarrow}}0.$

Hence $M_{4}PSH(\Omega)\subset M_{3}PSH(\Omega)$.

The proof is completed. ∎