Convergence in capacity
type: Item
instance of: scholarly article
title: Convergence in capacity
zbMATH Open document ID: 1152.32021
author: Yang Xing
publication date: 2008-08-26T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/10364
review text: The author studies how the convergence in capacity (for Bedford-Taylor relative capacity \(C_n\) and the capacity \(C_{n-1}\) defined by \[ C_{n-1} (E)=\sup \biggl\{ \int _E (dd^c u)^{n-1} \wedge (dd^c | z| ^2 ); \;u\in \text{PSH}(\Omega ),\;0<u<1 \biggr\} \] for compact \(E\subset \Omega\subset \mathbb C ^n \)) is related to the convergence of Monge-Ampère measures \((dd^c u_j )^n \to (dd^c u )^n\) associated to plurisubharmonic functions \(u_j\). He proves that if for \(u_j \in \text{PSH}(\Omega )\) \(u_j \to u\) and \((dd^c u_j )^n \to (dd^c u )^n\) weakly, and (essentially) \(u_j \geq u\) on \(\partial \Omega\) then \(u_j \to u\) in \(C_{n-1}\) capacity. He also obtains some generalizations of a stability theorem from \textit{U. Cegrell} and \textit{S. Kołodziej} [Math. Ann. 334, No.~4, 713--729 (2006; Zbl 1103.32019)] and a convergence theorem of \textit{U. Cegrell} [I. Newton Inst. for Math. Sci., preprint series NI01046-NPD].
Mathematics Subject Classification ID: 32W20
Mathematics Subject Classification ID: 32U15
zbMATH DE Number: 5315511
zbMATH Keywords: plurisubharmonic functions
zbMATH Keywords: Monge-Ampère operator
zbMATH Keywords: capacities
reviewed by: Slawomir Kołodziej
MaRDI profile type: Publication
OpenAlex ID: W1702004317
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DOI: 10.5802/AIF.2400
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 5315511