Estimates and structure of \(\alpha\)-harmonic functions
type: Item
instance of: scholarly article
title: Estimates and structure of \(\alpha\)-harmonic functions
zbMATH Open document ID: 1146.31004
author: M. Kwaśnicki
author: Krzysztof Bogdan
author: T. Kulczycki
publication date: 2008-04-03T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0607561
review text: The authors first prove a uniform boundary Harnack principle for nonnegative harmonic function for the fractional Laplacian in an arbitrary domain \(D\subset \mathbb{R}^d\), thus extending the result of \textit{R.~Song} and \textit{J.-M.~Wu} [J.~Funct.~Anal.~168, No. 2, 403--427 (1999; Zbl 0945.31006)], where the boundary Harnack principle was proved with the constant depending on the domain. Independence of the constant on the domain yields a unique representation of nonnegative harmonic functions as integrals against measures on \(D^c\cup \{\infty\}\) satisfying an integrability condition. The authors also identify the Martin boundary of \(D\) as the part of the Euclidean boundary consisting of the so called accessible points.
reviewed by: Zoran Vondraček
Mathematics Subject Classification ID: 31B05
Mathematics Subject Classification ID: 60J50
Mathematics Subject Classification ID: 31C35
Mathematics Subject Classification ID: 60G51
zbMATH DE Number: 5258616
zbMATH Keywords: boundary Harnack inequality
zbMATH Keywords: Martin representation
zbMATH Keywords: stable process
Wikidata QID: Q110598147
MaRDI profile type: Publication
OpenAlex ID: W2044540619
arXiv ID: math/0607561
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DOI: 10.1007/S00440-007-0067-0
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article