Equicontinuity of families of plurisubharmonic functions with bounds on their Monge-Ampère masses
type: Item
instance of: scholarly article
title: Equicontinuity of families of plurisubharmonic functions with bounds on their Monge-Ampère masses
zbMATH Open document ID: 1014.32023
author: Slawomir Kołodziej
published in: Mathematische Zeitschrift
publication date: 2002-12-01T00:00:00Z
review text: Let \(\varOmega\subset\mathbb C^n\) be a strictly pseudoconvex domain. The main results of the paper are the following two theorems. (1) Assume that \(\mathcal{PSH}(\varOmega)\ni u_j\longrightarrow u\in\mathcal{PSH}(\varOmega)\) weakly in \(\varOmega\), \(\liminf_{z\to\partial\varOmega}(u_j-u)(z)\geq 0\), and \((dd^cu_j)^n=f_jdV\), \(j=1,2,\dots\). The author characterizes a class of the functions \(f_j\) such that in the above situation \(u_j\longrightarrow u\) uniformly on \(\varOmega\). As a corollary he gets a result on the equicontinuity of certain families of solutions of the Monge-Ampère equation. (2) Let \(u, v\in\mathcal{PSH}(\varOmega)\), \(\lim_{z\to\partial\varOmega}(u-v)(z)\geq 0\), \((dd^cu)^n=fdV\), \((dd^cv)^n=gdV\) with \(\|f\|_p\), \(\|g\|_p\leq c_p\), \(p>1\), \(\frac 1p+\frac 1q=1\). Then for any \(\alpha\in(0,1)\) there exists \(c'=c'(p,n,\alpha,\varOmega)\) such that \(\|(u-v)_+\|_\infty\leq c'c_p^{\frac{\alpha+1}{n+\alpha n+1}} \|(u-v)_+\|_q^{\frac 1{n+\alpha n+1}}\).
reviewed by: Marek Jarnicki
Mathematics Subject Classification ID: 32U05
Mathematics Subject Classification ID: 32U40
zbMATH DE Number: 1837063
zbMATH Keywords: equicontinuity
zbMATH Keywords: families of solutions of the Monge-Ampère equation
MaRDI profile type: Publication
full work available at URL: s002090100401
OpenAlex ID: W2028134003
DOI: 10.1007/S002090100401
description: scientific article; zbMATH DE number 1837063