A Kuran type regularity criterion for sub-Laplacians. A necessary and sufficient condition
type: Item
instance of: scholarly article
title: A Kuran type regularity criterion for sub-Laplacians. A necessary and sufficient condition
zbMATH Open document ID: 1189.31008
author: Andrea Tommasoli
published in: Potential Analysis
publication date: 2010-01-29T00:00:00Z
review text: Let \(G\) be a stratified Lie group, \(\mathcal L\) its sub-Laplacian and \(\Gamma_x\) the fundamental solution of \(\mathcal L\) with pole at \(x.\) If \(\Omega\subset G\) is an open set, let \(\bar{H}_\varphi^\Omega\) and \(\underline{H}_\varphi^\Omega\) be the upper and the lower solution to the problem \(\mathcal Lu=0\) in \(\Omega\) and \(u|_{\partial\Omega}=\varphi\). If \(\overline r{H}_\varphi^\Omega=\underline{H}_\varphi^\Omega\) and they are \(\mathcal L\)-harmonic, then one writes \(H_\varphi^\Omega:=\overline{H}_\varphi^\Omega=\underline{H}_\varphi^\Omega.\) A point \(x\in\partial\Omega\) is called \(\mathcal L\)-regular if \(\lim_{\Omega\ni z\to x}H_\psi^\Omega=\psi(x)\) for every \(\psi\in C(\partial\Omega).\) A non-negative \(\mathcal L\)-harmonic function \(h\) in \(\Omega\) is said to be quasi-bounded in \(\Omega\) if there is an increasing sequence \(h_n\geq 0\) of bounded \(\mathcal L\)-harmonic functions in \(\Omega\) such that \(h=\lim_{n\to\infty}h_n\) in \(\Omega.\) The main theorem of the paper says that if \(\Omega\subset G\) is a bounded open set and \(x\in\partial\Omega\), then \(x\) is \(\mathcal L\)-regular if and only if \(\Gamma_x\) is quasi-bounded in \(\Omega.\)
reviewed by: Roman Urban
Mathematics Subject Classification ID: 31C05
Mathematics Subject Classification ID: 31B25
Mathematics Subject Classification ID: 35H20
zbMATH DE Number: 5664593
zbMATH Keywords: stratified Lie group
zbMATH Keywords: sub-Laplacian
zbMATH Keywords: \(\mathcal L\)-harmonic function
MaRDI profile type: Publication
full work available at URL: s11118-009-9144-6
OpenAlex ID: W1493058471
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cites work: Sur le rôle du point à l'infini dans la théorie des fonctions harmoniques
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cites work: Q5662251
cites work: A New Criterion of Dirichlet Regularity via the Quasi-Boundedness of the Fundamental Superharmonic Function
DOI: 10.1007/S11118-009-9144-6
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description: scientific article; zbMATH DE number 5664593