Some properties of the complex Monge-Ampere operator in Cegrell's classes and applications

TL;DR

This paper investigates convergence properties and decomposition theorems for the complex Monge-Ampère operator within Cegrell's classes, leading to a comparison principle applicable in complex analysis.

Contribution

It introduces a new convergence result in capacity and a general decomposition theorem for Monge-Ampère measures, enhancing understanding of the operator's properties.

Findings

Proved a convergence result in capacity for the Monge-Ampère operator.

Established a general decomposition theorem for Monge-Ampère measures.

Derived a comparison principle for the complex Monge-Ampère operator.

Abstract

In this article we will first prove a result about convergence in capacity. Using the achieved result we will obtain a general decompositon theorem for complex Monge-Ampere measues which will be used to prove a comparison principle for the complex Monge-Ampere operator.

Equations

$$C_n(E)=C_n(E,\Omega )=\sup \{\underset{E}{\int }(d{d}^{c}u{)}^n:u\in \text{PSH}(\Omega ),-1\le u\le 0\}$$

$$C_n(E)=\underset{E}{\int }(d{d}^c{h}_{E,\Omega }^{\ast }{)}^n$$

$$h_{E,\Omega }(z)=\sup \{u(z):u\in {\text{PSH}}^{-}(\Omega ),u\le -1\text{on}E\}.$$

$$\text{Cal}{E}_0=\text{Cal}{E}_0(\Omega )=\{\phi \in {\text{PSH}}^{-}(\Omega )\cap L^{\infty }(\Omega ):\underset{z\to \partial \Omega }{\lim }\phi (z)=0,\underset{\Omega }{\int }(d{d}^c\phi {)}^n<+\infty \},$$

$$\text{Cal}F=\text{Cal}F(\Omega )=\{\phi \in {\text{PSH}}^{-}(\Omega ):\exists \text{Cal}{E}_0(\Omega )\ni {\phi }_j\searrow \phi ,\underset{j\ge 1}{\sup }\underset{\Omega }{\int }(d{d}^c{\phi }_j{)}^n<+\infty \},$$

$$\text{Cal}E=\text{Cal}E(\Omega )=\{\phi \in {\text{PSH}}^{-}(\Omega ):\exists {\phi }_K\in \text{Cal}F(\Omega )\text{such that}{\phi }_K=\phi \text{on}K,\forall K\subset \subset \Omega \},$$

$$\text{Cal}{E}^a=\text{Cal}{E}^a(\Omega )=\{u\in \text{Cal}E(\Omega ):(d{d}^{c}u{)}^n(E)=0\forall E\text{is pluripolar in}\Omega \}.$$

$$e_0(u)=\underset{\Omega }{\int }(d{d}^{c}u{)}^n.$$

$$g(A)(z)=\sup \{u(z):u\in \text{Cal}{L}_A\}$$

$$\text{Cal}{L}_A=\{u\in {\text{PSH}}^{-}(\Omega ):u(z)-{\nu }_j\log |z-w_j|\le O(1)\text{as}z\to w_j,j=1,...,p\}$$

$$\nu (A)=\sum _{j=1}^p{\nu }_j^n,\hat{A}=\{w_j{\}}_{j=1,...,p}.$$

$$u(z)-v(z)\ge a-ϵ\text{for}z\in (\Omega \backslash K)\cap \{u>-\infty \}$$

$$v(z)=-\infty \text{for}z\in (\Omega \backslash K)\cap \{u=-\infty \}.$$

$$\frac{1}{(n!{)}^2}\underset{\{u<v\}}{\int }(v-u{)}^{n}d{d}^c{w}_1\wedge ...\wedge d{d}^c{w}_n+\underset{\{u<v\}}{\int }(r-w_1)(d{d}^{c}v{)}^n\le \underset{\{u<v\}}{\int }(r-w_1)(d{d}^{c}u{)}^n$$

$$\frac{1}{k!}\underset{\Omega }{\int }(v-u{)}^{k}d{d}^c{w}_1\wedge ...\wedge d{d}^c{w}_n+\underset{\Omega }{\int }(r-w_1)(d{d}^{c}v{)}^k\wedge d{d}^c{w}_{k+1}\wedge ...\wedge d{d}^c{w}_n$$

$$\le \underset{\Omega }{\int }(r-w_1)(d{d}^{c}u{)}^k\wedge d{d}^c{w}_{k+1}\wedge ...\wedge d{d}^c{w}_n$$

$$\frac{1}{k!}\underset{\Omega }{\int }(v-u{)}^{k}d{d}^c{w}_1\wedge ...\wedge d{d}^c{w}_n+\underset{\Omega }{\int }(r-w_1)(d{d}^{c}v{)}^k\wedge d{d}^c{w}_{k+1}\wedge ...\wedge d{d}^c{w}_n$$

$$\le \underset{\Omega }{\int }(r-w_1)(d{d}^{c}u{)}^k\wedge d{d}^c{w}_{k+1}\wedge ...\wedge d{d}^c{w}_n$$

$$\underset{\Omega }{\int }(v-u{)}^{k}d{d}^{c}w\wedge T\le k\underset{\Omega }{\int }(1-w)(v-u{)}^{k-1}d{d}^{c}u\wedge T$$

$$\underset{\Omega }{\int }(v-u{)}^{k}d{d}^{c}w\wedge T$$

$$=\underset{\Omega }{\int }(w-1)d{d}^c(v-u{)}^k\wedge T$$

$$=-k(k-1)\underset{\Omega }{\int }(1-w)d(v-u)\wedge d^c(v-u)\wedge T$$

$$+k\underset{\Omega }{\int }(1-w)(v-u{)}^{k-1}d{d}^c(u-v)\wedge T$$

$$\le k\underset{\Omega }{\int }(1-w)(v-u{)}^{k-1}d{d}^c(u-v)\wedge T$$

$$\le k\underset{\Omega }{\int }(1-w)(v-u{)}^{k-1}d{d}^{c}u\wedge T.$$

$$\underset{\Omega }{\int }(v_{ϵ}-u{)}^{k}d{d}^{c}w\wedge T\le k\underset{\Omega }{\int }(1-w)(v_{ϵ}-u{)}^{k-1}d{d}^{c}u\wedge T.$$

$$\underset{\Omega }{\int }(v-u{)}^{k}d{d}^{c}w\wedge T\le k\underset{\Omega }{\int }(1-w)(v-u{)}^{k-1}d{d}^{c}u\wedge T.$$

$$\frac{1}{k!}\underset{\Omega }{\int }(v-u{)}^{k}d{d}^c{w}_1\wedge ...\wedge d{d}^c{w}_n+\underset{\Omega }{\int }(r-w_1)(d{d}^{c}v{)}^k\wedge T$$

$$\le \underset{\Omega }{\int }(r-w_1)(d{d}^{c}u{)}^k\wedge T.$$

$$T=d{d}^c{w}_{k+1}\wedge ...\wedge d{d}^c{w}_n.$$

$$\underset{\Omega }{\int }(v-u{)}^{k}d{d}^c{w}_1\wedge ...\wedge d{d}^c{w}_n$$

$$\le ...$$

$$\le k!\underset{\Omega }{\int }(v-u)d{d}^c{w}_1\wedge (d{d}^{c}u{)}^{k-1}\wedge T$$

$$\le k!\underset{\Omega }{\int }(v-u)d{d}^c{w}_1\wedge [\sum _{i=0}^{k-1}(d{d}^{c}u{)}^i\wedge (d{d}^{c}v{)}^{k-i-1}]\wedge T$$

Full text

Some properties of the complex Monge-Ampère operator in

Cegrell’s classes and applications

NGUYEN VAN KHUE and PHAM HOANG HIEP

Abstract. In this article we will first prove a result about convergence in capacity. Using the achieved result we will obtain a general decompositon theorem for complex Monge-Ampère measues which will be used to prove a comparison principle for the complex Monge-Ampère operator. 2000 Mathematics Subject Classification: Primary 32W20, Secondary 32U15. Key words and phrases: complex Monge-Ampère operator, plurisubharmonic function. This work was supported by the National Research Program for Natural Sciences, Vietnam. 1. Introduction Let $\Omega$ be a bounded hyperconvex domain in $\mathbb{C}^{n}$. By PSH$(\Omega)$ we denote the set of plurisubharmonic (psh) functions on $\Omega$. In [BT 1,2] the authors established and used the comparison principle to study the Dirichlet problem in $\text{PSH}\cap L_{loc}^{\infty}(\Omega)$. Recently, Cegrell introduced a general class $\Cal{E}$ of psh functions on which the complex Monge-Ampère operator $(dd^{c}.)^{n}$ can be defined. He obtained many important results of pluripotential theory in the class $\Cal{E}$. For example, the ones on the comparison principle and solvability of the Dirichlet problem (see [Ce 1-3]). The main result of our paper are Theorem 4.1 and some Xing type comparision principles. Theorem 4.1 is generalize Lemma 5.4 in [Ce1], Lemma 7.2 in [Åh] and Lemma 3.4 in [Ce3]. For definitions of Cegrell’s classes see Section 2. After giving some preliminaries, we start in Proposition 3.1 with a comparison principle, which is analogous to a comparison principle due to Xing (Lemma 1 in [Xi1]). It should be observed that our proof is quite different from Xing’s proof, and the inequality we obtain is slightly stronger than Xing’s inequality, even in the case of bounded psh functions. Using Proposition 3.1, we give in Theorem 3.5 a sufficient condition for $C_{n}$-capacity convergence of a sequence of psh functions in the class $\Cal{F}$. This result should be compared to Theorem 3 of [Xi1] where the situation of bounded psh functions was studied. Applying Theorem 3.5 we give generalizations of recent results in [Cz] and [CLP] about convergences of multipole Green functions and a criterion for pluripolarity, respectively. Section 4 focuses on Theorem 4.1 and Theorem 4.9. By applying Theorem 4.1 we give some results on class Cegrell’s classes. We prove in Proposition 4.4 a local estimate for the Monge-Ampère measure in terms of the Beford-Taylor relative capacity. As an application, we give in Theorem 4.5 a decomposition result for Monge-Ampère measure, which is similar in spirit to Theorem 6.3 in [Ce1]. From Proposition 3.1 and Theorem 4.1 we obtain easily a Xing type comparison principle for functions in classes $\Cal{F}$ and $\Cal{E}$. Acknowledgment. We are grateful to Professor Urban Cegrell for useful discussions that helped to improve the paper. We are grateful to Per Åhag for fruitful comments. This work is supported by the National Research Program for Natural Sciences, Vietnam.

2. Preliminaries First we recall some elements of pluripotential theory that will be used throughout the paper. All this can be found in [BT2], [Ce1], [Ce2], [Le]. 2.1. We will always denote by $\Omega$ a bounded hyperconvex domain in $\mathbb{C}^{n}$ unless other wise stated. The $C_{n}$-capacity in the sense of Bedford and Taylor on $\Omega$ is the set function given by

[TABLE]

for every Borel set $E$ in $\Omega$. It is proved in [BT2] that

[TABLE]

where $h_{E,\Omega}^{*}$ is the upper regularization of the relative extremal function $h_{E,\Omega}$ for $E$ (relative to $\Omega$) i.e.,

[TABLE]

The following concepts are taken from [Xi1] and [Xi2] $*$A sequence of functions $u_{j}$ on $\Omega$ is said to converge to a function $u$ in $C_{n}$-capacity on a set $E\subset\Omega$ if for every $\delta>0$ we have $C_{n}(\{z\in E\ :\ |u_{j}(z)-u(z)|>\delta\})\to 0$ as $j\to\infty.$ $*$A family of positive measures {$\mu_{\alpha}$} on $\Omega$ is called uniformly absolutely continuous with respect to $C_{n}$-capacity in a set $E\subset\Omega$ if for every $\epsilon>0$ there exists $\delta>0$ such that for each Borel subset $F\subset E$ with $C_{n}$(F)$<\delta$ the inequality $\mu_{\alpha}$(F)$<\epsilon$ holds for all $\alpha$. We write $\mu_{\alpha}\ll C_{n}$ in $E$ uniformly for $\alpha$. 2.2. The following classes of psh functions were introduced by Cegrell in [Ce1] and [Ce2]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

For each $u\in\Cal{F}(\Omega)$, we set

[TABLE]

2.3. Let $A=\{(w_{j},\nu_{j})\}_{j=1,...,p}$ be a finite subset of $\Omega\times\mathbb{R}^{+}$. According to Lelong (see [Le]), the pluricomplex Green function with poles in $A$ is defined by

[TABLE]

where

[TABLE]

Set

[TABLE]

2.4. We write $\varliminf\limits_{z\to\partial\Omega}[u(z)-v(z)]\geq a$ if for every $\epsilon>0$ there exists a compact set K in $\Omega$ such that

[TABLE]

and

[TABLE]

2.5. Xing’s comparison principle (see Lemma 1 in [Xi1]). Let $\Omega$ be a bounded open subset in $\mathbb{C}^{n}$ and $u,v\in\text{PSH}\cap L^{\infty}(\Omega)$ satisfy $\varliminf\limits_{z\to\partial\Omega}[u(z)-v(z)]\geq 0$. Then for any constant $r\geq 1$ and all $w_{j}\in\text{PSH}(\Omega)$ with $0\leq w_{j}\leq 1$, $j=1,2,...,n$ we have

[TABLE]

3. Some convergence theorems In order to study the convergence of a sequence of psh functions in $C_{n}$-capacity, we start with the following. 3.1. Proposition. a) Let $u,v\in\Cal{F}$such that $u\leq v$ on $\Omega$. Then for $1\leq k\leq n$

[TABLE]

[TABLE]

for all $w_{j}\in\text{PSH}(\Omega),\ 0\leq w_{j}\leq 1,\ j=1,...,k,\ w_{k+1},...,w_{n}\in\Cal{F}$ and all $r\geq 1$. b) Let $u,v\in\Cal{E}$ such that $u\leq v$ on $\Omega$ and $u=v$ on $\Omega\backslash K$ for some $K\subset\subset\Omega$. Then for $1\leq k\leq n$

[TABLE]

[TABLE]

for all $w_{j}\in\text{PSH}(\Omega),\ 0\leq w_{j}\leq 1,\ j=1,...,k,\ w_{k+1},...,w_{n}\in\Cal{E}$ and all $r\geq 1$. We proceed through some lemmas. 3.2. Lemma. Let $u,v\in\text{PSH}\cap L^{\infty}(\Omega)$ such that $u\leq v$ on $\Omega$ and $\lim\limits_{z\to\partial\Omega}[u(z)-v(z)]=0$. Then

[TABLE]

for all $w\in\text{PSH}(\Omega)$, $0\leq w\leq 1$ and all positive closed currents $T$. Proof. First, assume $u,v\in\text{PSH}\cap L^{\infty}(\Omega)$, $u\leq v$ on $\Omega$ and $u=v$ on $\Omega\backslash K$, $K\subset\subset\Omega$. Then, using the Stokes formula we obtain

[TABLE]

General case, for each $\epsilon>0$ we set $v_{\epsilon}=\max(u,v-\epsilon)$. Then $v_{\epsilon}\nearrow v$ on $\Omega$, $v_{\epsilon}\geq u$ on $\Omega$ and $v_{\epsilon}=u$ on $\Omega\backslash K$ for some $K\subset\subset\Omega$. Hence

[TABLE]

Since $0\leq v_{\epsilon}-u\nearrow v-u$ as $\epsilon\searrow 0$, letting $\epsilon\searrow 0$ we get

[TABLE]

3.3. Lemma. Let $u,v\in\text{PSH}\cap L^{\infty}(\Omega)$ such that $u\leq v$ on $\Omega$ and $\lim\limits_{z\to\partial\Omega}[u(z)-v(z)]=0$. Then for $1\leq k\leq n$

[TABLE]

[TABLE]

for all $w_{1},...,w_{k}\in\text{PSH}(\Omega),\ 0\leq w_{j}\leq 1\ \forall\ j=1,...,k,\ w_{k+1},...,w_{n}\in\Cal{E}$ and all $r\geq 1$. Proof. To simplify the notation we set

[TABLE]

First, assume that $u,v\in\text{PSH}\cap L^{\infty}(\Omega),\ u\leq v$ on $\Omega$, and $u=v$ on $\Omega\backslash K$, $K\subset\subset\Omega$. Using Lemma 3.2 we get

[TABLE]

General case, for each $\epsilon>0$ we put $v_{\epsilon}=\max(u,v-\epsilon)$. Then $v_{\epsilon}\nearrow v$ on $\Omega$, $v_{\epsilon}\geq u$ on $\Omega$ and $v_{\epsilon}=u$ on $\Omega\backslash K$ for some $K\subset\subset\Omega$. Hence

[TABLE]

[TABLE]

Observe that $0\leq v_{\epsilon}-u\nearrow v-u$ and $(dd^{c}v_{\epsilon})^{k}\wedge T\to(dd^{c}v)^{k}\wedge T$ weakly as $\epsilon\searrow 0$, $r-w_{1}$ is lower semicontinuous, by letting $\epsilon\searrow 0$ we have

[TABLE]

[TABLE]

The proof is finished. Proof of Proposition 3.1. a) Let $\Cal{E}_{0}\ni u_{j}\searrow u$ and $\Cal{E}_{0}\ni v_{j}\searrow v$ as in the definition of $\Cal{F}$. Replace $v_{j}$ by $\max(u_{j},v_{j})$ we may assume that $u_{j}\leq v_{j}$ for $j\geq 1$. By Lemma 3.3 we have

[TABLE]

[TABLE]

for $t\geq j\geq 1$. By Proposition 5.1 in [Ce2] letting $t\to\infty$ in the above inequality we have

[TABLE]

[TABLE]

for $j\geq 1$. Next letting $j\to\infty$ again by Proposition 5.1 in [Ce2] we get the desired conclusion. b) Let $G,W$ be open sets such that $K\subset\subset G\subset\subset W\subset\subset\Omega$. According to the remark following Definition 4.6 in [Ce2] we can choose a function $\tilde{v}\in\Cal{F}$ such that $\tilde{v}\geq v$ and $\tilde{v}=v$ on $W$. Set

[TABLE]

Since $u=v=\tilde{v}$ on $W\backslash K$ we have $\tilde{u}\in\text{PSH}^{-}(\Omega)$. It is easy to see that $\tilde{u}\in\Cal{F}$, $\tilde{u}\leq\tilde{v}$ and $\tilde{u}=u$ on $W$. By a) we have

[TABLE]

[TABLE]

Since $\tilde{u}=\tilde{v}$ on $\Omega\backslash G$ we have

[TABLE]

[TABLE]

Since $\tilde{u}=u$, $\tilde{v}=v$ on $W$ and $u=v$ on $\Omega\backslash K$ we obtain

[TABLE]

[TABLE]

3.4. Proposition. Let $u,v\in\Cal{F}$ and $u\leq v$ on $\Omega$. Then

[TABLE]

for all $w_{j}\in\text{PSH}(\Omega)$, $-1\leq w_{j}\leq 0$, $j=1,...,n$. Proof. The proposition follows from Proposition 3.1 with $k=n,\ r=1$ and $w_{j}$ are replaced by $w_{j}+1$. 3.5. Theorem. Let $u,u_{j}\in\Cal{F}$ and $u_{j}\leq u$ for $j\geq 1$. Assume that $\sup\limits_{j\geq 1}\int\limits_{\Omega}(dd^{c}u_{j})^{n}<+\infty$ and $||(dd^{c}u_{j})^{n}-(dd^{c}u)^{n}||_{E}\to 0$ as $j\to\infty$ for all $E\subset\subset\Omega$. Then $u_{j}\to u$ in $C_{n}$-capacity on every $E\subset\subset\Omega$ as $j\to\infty$. Proof. Let $\Omega^{\prime}\subset\subset\Omega$ and $\delta>0$. Put

[TABLE]

We prove that $C_{n}(A_{j})\to 0$ as $j\to\infty$. Given $\epsilon>0$. By quasicontinuity of $u$ and $u_{j}$, there is an open set $G$ in $\Omega$ such that $C_{n}(G)<\epsilon$, and $u_{j}|_{\Omega\backslash G},\ u|_{\Omega\backslash G}$ are continuous. We have

[TABLE]

where $B_{j}=\{z\in\overline{\Omega^{\prime}}\backslash G\ :\ u-u_{j}\geq\delta\}$ are compact sets in $\Omega$ and

[TABLE]

We claim that $\lim\limits_{j\to\infty}C_{n}(B_{j})=0$. By Proposition 3.4 we have

[TABLE]

As $\lim\limits_{z\to\partial\Omega}h_{\Omega^{\prime}}(z)=0$ there exists $K\subset\subset\Omega$ such that

[TABLE]

By the hypothesis

[TABLE]

Thus

[TABLE]

This proves the claim and hence the theorem. As an application of Theorem 3.5 we have the following 3.6. Proposition. Let $g(A_{j})$ be multipolar Green functions on $\Omega$ such that

[TABLE]

Then $g(A_{j})\to 0$ as $j\to\infty$ in $C_{n}$-capacity. Proof. By the hypothesis we have

[TABLE]

and

[TABLE]

Theorem 3.5 implies that $g(A_{j})\to 0$ as $j\to\infty$ in $C_{n}$-capacity. This section ends up with a criterion for pluripolarity 3.7. Theorem. Let $u_{j}\in\Cal{F}$ such that $\sup\limits_{j\geq 1}\int\limits_{\Omega}(dd^{c}u_{j})^{n}<+\infty$. Then there is a constant $A>0$ such that

i)$(\varlimsup\limits_{j\to\infty}u_{j})^{*}\in\Cal{F}.$ ii)$C_{n}(\{z\in\Omega:\ (\varlimsup\limits_{j\to\infty}u_{j})^{*}(z)<-t\})\leq\frac{A}{t^{n}}.$ iii)$\{z\in\Omega:\ \lim\limits_{j\to\infty}u_{j}(z)=-\infty\}$ is pluripolar. Proof. i) For each $j\geq 1$ put $v_{j}=\sup\{u_{j},u_{j+1},...\}.$ By [Ce2] $v_{j}^{*}\in\Cal{F}$ and

[TABLE]

By [Ce2] we have $v_{j}^{*}\searrow v\in\Cal{F}$. ii) By Proposition 3.1 in [CKZ] we have

[TABLE]

where $A=2^{n}e_{0}(v).$ iii) According to [BT2] we have

[TABLE]

Remark. Theorem 3.7 in the case where $u_{j}$ are multipole Green functions was proved by D.Coman, N.Levenberg and A.Poletsky in Theorem 4.1 of [CLP].

4. Some properties of the Cegrell’s classes and applications In this section, first we prove the following 4.1. Theorem. Let $u,u_{1},...,u_{n-1}\in\Cal{E}$, $v\in\text{PSH}^{-}(\Omega)$ and $T=dd^{c}u_{1}\wedge...\wedge dd^{c}u_{n-1}$. Then

[TABLE]

We need the following well-known fact. 4.2. Lemma. Let $\mu$ be a measure on $\Omega$ and $f:\Omega\to\mathbb{R}$ a measurable function on $\Omega$. The following are equivalent i)$\mu(E)=0$ for all Borell sets $E\subset\{f\neq 0\}.$ ii)$\int\limits_{E}fd\mu=0$ for every measurable set $E$ in $\Omega$. Proof. i)$\Rightarrow$ii) follows from:

[TABLE]

ii)$\Rightarrow$i). It suffices to show that $\mu=0$ on every $X_{\delta}=\{f>\delta>0\}$. By the Hahn decomposition theorem, there exist measurable subsets $X_{\delta}^{+}$ and $X_{\delta}^{-}$ of $X_{\delta}$ such that $X_{\delta}=X_{\delta}^{+}\cup X_{\delta}^{-}$, $X_{\delta}^{+}\cap X_{\delta}^{-}=\emptyset$ and $\mu\geq 0$ on $X_{\delta}^{+}$, $\mu\leq 0$ on $X_{\delta}^{-}$. We have

[TABLE]

Hence, $\mu(X_{\delta}^{+})=\mu(X_{\delta}^{-})=0$. Therefore, we have $\mu=0$ on $X_{\delta}$. Proof of Theorem 4.1. a) First we prove the proposition for $v\equiv a<0$. According to the remark following Definition 4.6 in [Ce2], without loss of generality we may assume that $u,u_{1},...,u_{n-1}\in\Cal{F}$. Using Theorem 2.1 in [Ce2] we can find

[TABLE]

Since $\{u^{j}>a\}$ is open we have

[TABLE]

Thus from the inclusion $\{u>a\}\subset\{u^{j}>a\}$ we obtain

[TABLE]

where $T_{j}=dd^{c}u_{1}^{j}\wedge...\wedge dd^{c}u_{n-1}^{j}.$ By Corollary 5.2 in [Ce2], it follows that

[TABLE]

[TABLE]

Hence

[TABLE]

Using Lemma 4.2 we have

[TABLE]

b) Assume that $v\in PSH^{-}(\Omega)$. Since $\{u>v\}=\bigcup\limits_{a\in\mathbb{Q}^{-}}\{u>a>v\}$, it suffices to show that

[TABLE]

for all $a\in\mathbb{Q}^{-}$. Since $\max(u,v)\in\Cal{E}$, by a) we have

[TABLE]

[TABLE]

Since $\max(u,v,a)=\max(u,a)$ on set open $\{a>v\}$ , we have

[TABLE]

Since $\{u>a>v\}\subset\{u>a\},\ \{a>v\},\ \{\max(u,v)>a\}$ and (1), (2), (3) we have

[TABLE]

The next result is an analogue of an inequality due to Demaily in [De2] 4.3. Proposition. a) $u,v\in\Cal{E}$ such that $(dd^{c}u)^{n}(\{u=v=-\infty\})=0$. Then

[TABLE]

where $1_{E}$ denotes the characteristic function of $E$. b) Let $\mu$ be a positive measure which vanishes on all pluripolar subsets of $\Omega$. Suppose $u,v\in\Cal{E}$ such that $(dd^{c}u)^{n}\geq\mu,(dd^{c}v)^{n}\geq\mu$. Then $(dd^{c}\max(u,v))^{n}\geq\mu$. Proof. a) For each $\epsilon>0$ put $A_{\epsilon}=\{u=v-\epsilon\}\backslash\{u=v=-\infty\}$. Since $A_{\epsilon}\cap A_{\delta}=\emptyset$ for $\epsilon\neq\delta$ there exists $\epsilon_{j}\searrow 0$ such that $(dd^{c}u)^{n}(A_{\epsilon_{j}})=0$ for $j\geq 1$. On the other hand, since $(dd^{c}u)^{n}(\{u=v=-\infty\})=0$ we have $(dd^{c}u)^{n}(\{u=v-\epsilon_{j}\})=0$ for $j\geq 1$. Since Theorem 4.1 it follows that

[TABLE]

Letting $j\to\infty$ and by Remark under Theorem 5.15 in [Ce2] we get

[TABLE]

because $\max(u,v-\epsilon_{j})\nearrow\max(u,v)$ and $1_{\{u<v-\epsilon_{j}\}}\nearrow 1_{\{u<v\}}$ as $j\to\infty$. b) Argument as a) 4.4. Proposition. Let $u_{1},...,u_{k}\in\text{PSH}(\Omega)\cap L^{\infty}(\Omega)$ and $u_{k+1},...,u_{n}\in\Cal{E}$. Then i) $\int\limits_{B}dd^{c}u_{1}\wedge...\wedge dd^{c}u_{n}=O((C_{n}(B))^{\frac{k}{n}})$ for all Borel sets $B\subset\Omega^{\prime}\subset\subset\Omega.$ ii) $\int\limits_{B(a,r)}dd^{c}u_{1}\wedge...\wedge dd^{c}u_{n}=o((C_{n}(B(a,r)))^{\frac{k}{n}})$ as $r\to 0$ for all $a\in\Omega.$ where $B(a,r)=\{z\i\mathbb{C}^{n}\ :\ |z-a|<r\}$ Proof. We may assume that $0\leq u_{j}\leq 1$ for $j=1,...,k$. On the other hand, by the remark following Defintion 4.6 in [Ce2] we again may assume that $u_{k+1},...,u_{n}\in\Cal{F}.$ i) For each open set $B\subset\subset\Omega$, applying Proposition 3.1 we get

[TABLE]

Hence

[TABLE]

for all Borel set $B\subset\Omega$. ii) By Proposition 3.1 we have

[TABLE]

Hence $(-\varphi)^{k}\in L_{1}(dd^{c}u_{1}\wedge...\wedge dd^{c}u_{n})$ for all $\varphi\in\Cal{F}(\Omega)$. Given $a\in\Omega$ let $r_{0},R_{0}$ such that $B(a,r_{0})\subset\subset\Omega\subset\subset B(a,R_{0})$. Then

[TABLE]

for all $z\in\Omega$, where $g_{a}$ denotes the Green function of $\Omega$ with pole at $a$. Since $(-g_{a})^{k}\in L_{1}(dd^{c}u_{1}\wedge...\wedge dd^{c}u_{n})$, it follows that

[TABLE]

Hence

[TABLE]

as $r\to 0$. This means that

[TABLE]

Combining this with the inequality

[TABLE]

we get

[TABLE]

The next result should be compared with Theorem 6.3 in [Ce1] 4.5. Theorem. Let $u_{1},...,u_{n}\in\Cal{E}$. Then there exists $\tilde{u}\in\Cal{E}^{a}$ such that

[TABLE]

Proof. First, we write

[TABLE]

where

[TABLE]

It is easy to see that $\mu\ll C_{n}$ in every $E\subset\subset\Omega$. Indeed, by Theorem 4.1 we have

[TABLE]

Hence, by Proposition 4.4 (i) it follows that $dd^{c}u_{1}\wedge...\wedge dd^{c}u_{n}|_{\{u_{1}>-j\}}\ll C_{n}$ in every $E\subset\subset\Omega$. Next, it remains to show that there exists $\tilde{u}\in\Cal{E}^{a}$ such that $\mu=(dd^{c}\tilde{u})^{n}$. Let $\{\Omega_{j}\}$ be an increasing exhaustion sequence of $\Omega$. For each $j\geq 1$ put $\mu_{j}=\mu|_{\Omega_{j}}$. By [Åh] there exists $\tilde{u}_{j}\in\Cal{F}$ such that $(dd^{c}\tilde{u}_{j})^{n}=\mu_{j}$. Notice that $\mu_{j}\nearrow\mu$ and

[TABLE]

Applying the comparison principle we obtain

[TABLE]

Hence, $\tilde{u}\in\Cal{E}^{a}$ and $(dd^{c}\tilde{u})^{n}=\lim\limits_{j\to\infty}(dd^{c}\tilde{u}_{j})^{n}=\mu$. The proof is thereby completed. 4.6. Corollary. $u_{1},...,u_{n}\in\Cal{E}$. Then the following are equivalent i) $dd^{c}u_{1}\wedge...\wedge dd^{c}u_{n}\ll C_{n}$ in every $E\subset\subset\Omega.$ ii) $\int\limits_{\{u_{1}=...=u_{n}=-\infty\}}dd^{c}u_{1}\wedge...\wedge dd^{c}u_{n}=0.$ iii) $\int\limits_{\{u_{1}<-s,...,u_{n}<-s\}\cap E}dd^{c}u_{1}\wedge...\wedge dd^{c}u_{n}\to 0$ as $s\to+\infty$ for all $E\subset\subset\Omega$. Proof. Direct application of Theorem 4.5. The comparison principle for class $\Cal{F}$ was studied in [Ce3] and [H1]. By using Proposition 3.1 and Theorem 4.1 we prove a Xing type comparison principle for $\Cal{F}$ 4.7. Theorem. Let $u\in\Cal{F}$, $v\in\Cal{E}$ and $1\leq k\leq n$. Then

[TABLE]

[TABLE]

for all $w_{j}\in\text{PSH}(\Omega),\ 0\leq w_{j}\leq 1,\ j=1,...,k,\ w_{k+1},...,w_{n}\in\Cal{F}$ and all $r\geq 1$. Proof. Let $\epsilon>0$. We set $\tilde{v}=\max(u,v-\epsilon)$. By a) in Proposition 3.1 we have

[TABLE]

[TABLE]

Since $\{u<\tilde{v}\}=\{u<v-\epsilon\}$ and Theorem 4.1 we have

[TABLE]

[TABLE]

[TABLE]

Letting $\epsilon\searrow 0$ we obtain

[TABLE]

[TABLE]

4.8. Corollary. Let $u\in\Cal{E}^{a}$ such that $u\geq v$ for all functions $v\in\Cal{E}$ satisfying $(dd^{c}u)^{n}\leq(dd^{c}v)^{n}$. Then

[TABLE]

[TABLE]

for all $v\in\Cal{E}$, $r\geq 1$ and all $w_{1},...,w_{n}\in\text{PSH}(\Omega)$, $0\leq w_{1},...,w_{n}\leq 1$. Proof. Let $\{\Omega_{j}\}$ be an increasing exhaustion sequence of relatively compact subdomains of $\Omega$. Set $\mu_{j}=1_{\Omega_{j}}1_{\{u>-j\}}(dd^{c}u)^{n}$, where $1_{E}$ denotes the characteristic function of $E\subset\Omega$. Applying Theorem 4.1 we have

[TABLE]

Take $\phi\in\Cal{E}_{0}(\Omega)\cap C(\bar{\Omega})$. Put

[TABLE]

where $a_{j}=\frac{-j}{\sup\limits_{\Omega_{j+1}}\phi}.$ Then $\phi_{j}=\max(u,-j)$ on $\Omega_{j+1}$, $\phi_{j}\in\Cal{E}_{0}$ and

[TABLE]

By Kołodziej’s theorem (see [Ko]) there exists $u_{j}\in\Cal{E}_{0}$ such that

[TABLE]

for all $j\geq 1$. By the comparison principle we have $u_{j}\searrow\tilde{u}\geq u$. On the other hand, since $(dd^{c}u)^{n}(\{u=-\infty\})=0$, it follows that

[TABLE]

weakly as $j\to\infty.$ Thus $(dd^{c}\tilde{u})^{n}=\lim\limits_{j\to\infty}(dd^{c}u_{j})^{n}=(dd^{c}u)^{n}$. By the hypothesis we have $\tilde{u}=u$. Applying Theorem 4.7 we get

[TABLE]

[TABLE]

[TABLE]

Letting $j\to\infty$ we obtain

[TABLE]

Arguing as in Theorem 4.7 we prove a Xing type comparison principle for $\Cal{E}$. 4.9. Theorem. Let $u,v\in\Cal{E}$ and $1\leq k\leq n$ such that $\varliminf\limits_{z\to\partial\Omega}[u(z)-v(z)]\geq 0$. Then

[TABLE]

[TABLE]

for all $w_{j}\in\text{PSH}(\Omega),\ 0\leq w_{j}\leq 1,\ j=1,...,k,\ w_{k+1},...,w_{n}\in\Cal{E}$ and all $r\geq 1$. Proof. Let $\epsilon>0$. We set $\tilde{v}=\max(u,v-\epsilon)$. By b) in Proposition 3.1 we have

[TABLE]

[TABLE]

Since $\{u<\tilde{v}\}=\{u<v-\epsilon\}$ and Theorem 4.1 we have

[TABLE]

[TABLE]

[TABLE]

Letting $\epsilon\searrow 0$ we obtain

[TABLE]

[TABLE]

References

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Department of Mathematics Hanoi University of Education (Dai hoc Su Pham Hanoi). Cau giay, Ha Noi, VietNam E-mail: phhiep-vnyahoo.com