A note on maximal subextensions of plurisubharmonic functions
type: Item
instance of: scholarly article
title: A note on maximal subextensions of plurisubharmonic functions
zbMATH Open document ID: 1394.32026
author: Hai Mau Le
author: Tran Van Thuy
author: Nguyen Xuan Hong
published in: Acta Mathematica Vietnamica
publication date: 2018-02-14T00:00:00Z
review text: The authors study the convergence in capacity of maximal subextensions of plurisubharmonic functions from the Cegrell's class \(\mathcal E\). Let \(\Omega\subset\widehat\Omega\subset\subset\mathbb C^n\) be hyperconvex domains and let \(u\in\mathcal{PSH}(\Omega)\). A function \(\widehat u\in\mathcal{PSH}(\widehat\Omega)\) is called a subextension of \(u\) if \(\widehat u|_{\Omega}\leq u\). For \(f\in\mathcal E(\Omega)\) let \(\mathcal F^a(\Omega,f)\) denote the family of all \(u\in\mathcal{PSH}(\Omega)\) such that \(\psi+f\leq u\leq f\) for a function \(\psi\in\mathcal{PSH}(\Omega)\) such that \((dd^c\psi)^n=0\) on all pluripolar subsets of \(\Omega\). Let \(g\in\mathcal E(\widehat\Omega)\), \(g|_\Omega\leq f\), and \(u\in\mathcal F(\Omega,f)\). The function \(S_{u,g}:=\sup\{\varphi\in\mathcal{PSH}(\widehat\Omega): \varphi\leq g,\;\varphi|_\Omega\leq u\}\) is called the maximal subextension of \(u\) with boundary value \(g\). The main result of the paper is the following theorem. Let \(w\in\mathcal F^a(\Omega,f)\) be such that \(\int_{\widehat\Omega}(dd^cg)^n+\int_\Omega(dd^cw)^n<+\infty\). Assume that \((u_j)_{j=1}^\infty\subset\mathcal F^a(\Omega,f)\) is such that \(u_j\geq w\), \(j\in\mathbb N\), and \(u_j\longrightarrow u\) in capacity. Then \(S_{u_j,g}\longrightarrow S_{u,g}\) in capacity.
reviewed by: Marek Jarnicki
Mathematics Subject Classification ID: 32U05
Mathematics Subject Classification ID: 32U15
Mathematics Subject Classification ID: 32W20
zbMATH DE Number: 6838516
zbMATH Keywords: plurisubharmonic functions
zbMATH Keywords: subextension of plurisubharmonic functions
zbMATH Keywords: convergence in capacity
MaRDI profile type: Publication
full work available at URL: s40306-017-0234-z
OpenAlex ID: W2765843693
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DOI: 10.1007/S40306-017-0234-Z
description: scientific article; zbMATH DE number 6838516