Approximation of maximal plurisubharmonic functions

TL;DR

This paper investigates the approximation of maximal plurisubharmonic functions in complex analysis, establishing new convergence properties for sequences of such functions and generalizing existing approximation results.

Contribution

It introduces a novel convergence criterion involving weighted Monge-Ampère measures for sequences approximating maximal plurisubharmonic functions.

Findings

Weighted measures $(|u_j|+1)^{-a} (dd^c u_j)^n$ converge to zero for $a>n-1$

Generalizes classical approximation results for maximal plurisubharmonic functions

Provides new insights into the behavior of Monge-Ampère measures in approximation sequences

Abstract

Let $u$ be a maximal plurisubharmonic function in a domain $\Omega\subset\mathbb{C}^n$ ($n\geq 2$). It is classical that, for any $U\Subset\Omega$, there exists a sequence of bounded plurisubharmonic functions $PSH(U)\ni u_j\searrow u$ satisfying the property: $(dd^c u_j)^n$ is weakly convergent to $0$ as $j\rightarrow\infty$. In general, this property does not hold for arbitrary sequence. In this paper, we show that for any sequence of bounded plurisubharmonic functions $PSH(U)\ni u_j\searrow u$, $(|u_j|+1)^{-a} (dd^cu_j)^n$ is weakly convergent to $0$ as $j\rightarrow\infty$, where $a>n-1$. We also generalize some well-known results about approximation of maximal plurisubharmonic functions.