Fatou theorem of \(p\)-harmonic functions on trees.
type: Item
instance of: scholarly article
title: Fatou theorem of \(p\)-harmonic functions on trees.
zbMATH Open document ID: 1038.31007
published in: The Annals of Probability
publication date: 2003-05-06T00:00:00Z
review text: The authors study bounded \(p\)-harmonic functions \(u\) defined on a directed tree \(T\) with branching order \(\kappa\) (\(1<p<\infty\) and \(\kappa=2,3,\dots\)). Denote by \(BV(u)\) the set of paths on which \(u\) has finite variation and \(\mathcal{F}(u)\) the set of paths on which \(u\) has a finite limit. Then the infimum of dim \(BV(u)\) and the infimum of dim \(\mathcal{F}(u)\) are equal over all bounded \(p\)-harmonic functions on \(T\) (with \(p\) and \(\kappa\) fixed); the infimum \(d(\kappa,p)\) is attained and is strictly between 0 and 1 except when \(p=2\) or \(\kappa=2\).
Mathematics Subject Classification ID: 31C20
Mathematics Subject Classification ID: 31C45
Mathematics Subject Classification ID: 31A20
Mathematics Subject Classification ID: 60G42
zbMATH DE Number: 1905943
zbMATH Keywords: Fatou theorem
zbMATH Keywords: \(p\)-harmonic functions
zbMATH Keywords: directed tree
zbMATH Keywords: finite variation
zbMATH Keywords: finite limit
author: Robert Kaufman
author: Jang-Mei Wu
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1019160328
OpenAlex ID: W2009185127
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DOI: 10.1214/AOP/1019160328
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description: scientific article; zbMATH DE number 1905943