A note on Dirichlet regularity on harmonic spaces
type: Item
instance of: scholarly article
title: A note on Dirichlet regularity on harmonic spaces
zbMATH Open document ID: 0733.31007
author: Noriaki Suzuki
published in: Hiroshima Mathematical Journal
publication date: 1991-01-01T00:00:00Z
review text: Let X be a locally compact Hausdorff space with a countable base. A harmonic space (X,U) is said to have an adjoint structure \(U^*\) if both (X,U) and \((X,U^*)\) are P-harmonic spaces in the sense of \textit{C. Constantinescu} and \textit{A. Cornea} [Potential theory on harmonic spaces (1972; Zbl 0248.31011)] and the associated Green function G(x,y) satisfies some auxiliary conditions. The following criterion for the regularity of the boundary point is proved. Let D be an open set in X and let \(x_ 0\in \partial D\). If \(x_ 0\) is a U-regular boundary point, then \(G(x_ 0,\cdot)\) is \(U^*\)-quasi- bounded on D. The converse assertion is valid if either \(\{x_ 0\}\) is a \(U^*\)-polar set or \(G(x_ 0,\cdot)\) is continuous on \(\bar D.\) The corresponding criterium for the parabolic periodic \(\Delta\)-\(\partial /\partial t\) is given as a corollary.
Mathematics Subject Classification ID: 31B35
Mathematics Subject Classification ID: 31C05
Mathematics Subject Classification ID: 31D05
zbMATH DE Number: 4212423
zbMATH Keywords: harmonic space
zbMATH Keywords: Green function
zbMATH Keywords: regularity of the boundary point
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.32917/hmj/1206130970
OpenAlex ID: W1596461291
Wikidata QID: Q128871422
DOI: 10.32917/HMJ/1206130970
description: scientific article; zbMATH DE number 4212423