Remarks on weak convergence of complex Monge-Amp\`ere measures

TL;DR

This paper investigates conditions under which weak convergence of Monge-Ampère measures occurs for decreasing sequences of plurisubharmonic functions, focusing on the nonpolar part and the inclusion of the limit function in the domain.

Contribution

It provides new criteria ensuring the weak convergence of Monge-Ampère measures and characterizes when the limit function belongs to the domain of the Monge-Ampère operator.

Findings

Established conditions for measure convergence involving the nonpolar part.

Linked measure convergence to the limit function's domain inclusion.

Extended results to cases with maximum functions of the sequence.

Abstract

Let $(u_j)$ be a deaceasing sequence of psh functions in the domain of definition $\cal D$ of the Monge-Amp\`ere operator on a domain $\Omega$ of $\mathbb{C}^n$ such that $u=\inf_j u_j$ is plurisubharmonic on $\Omega$. In this paper we are interested in the problem of finding conditions insuring that \begin{equation*} \lim_{j\to +\infty} \int\varphi (dd^cu_j)^n=\int\varphi {\rm NP}(dd^cu)^n \end{equation*} for any continuous function on $\Omega$ with compact support, where ${\rm NP}(dd^cu)^n$ is the nonpolar part of $(dd^cu)^n$, and conditions implying that $u\in \cal D$. For $u_j=\max(u,-j)$ these conditions imply also that \begin{equation*} \lim_{j\to +\infty} \int_K(dd^cu_j)^n=\int_K {\rm NP}(dd^cu)^n \end{equation*} for any compact set $K\subset\{u>-\infty\}$.