Sobolev space properties of superharmonic functions on metric spaces
type: Item
instance of: scholarly article
title: Sobolev space properties of superharmonic functions on metric spaces
zbMATH Open document ID: 1042.31005
DOI: 10.1007/BF03322918
author: Juha Kinnunen
author: O. Martio
published in: Results in Mathematics
publication date: 2004-03-25T00:00:00Z
review text: In the recent years, after the publication of the important paper by \textit{J. Heinonen} and \textit{P. Koskela} [Acta Math. 181, No. 1, 1--61 (1998; Zbl 0915.30018)], there has been a period of intensive study of nonlinear analysis in metric spaces. The leading idea is to generalize, as much as possible, nonlinear potential theory and quasiconformal mapping theory to the setup of spaces that are not even locally euclidean, however, under some local homogeneity postulates such as the existence of a doubling Borel measure or the validity of some integral inequalities. This development of the past few years shows that starting with a few postulates much of the classical theory of superharmonic functions extends to this general setup. The authors prove a local integrability result for superharmonic functions generalizing a theorem of \textit{P. Lindqvist} [J. Reine Angew. Math. 365, 67--79 (1986; Zbl 0572.31004)].
Mathematics Subject Classification ID: 31C45
Mathematics Subject Classification ID: 46E35
Mathematics Subject Classification ID: 31C05
Mathematics Subject Classification ID: 54E35
zbMATH DE Number: 2062006
zbMATH Keywords: Sobolev spaces
zbMATH Keywords: metric spaces
zbMATH Keywords: superharmonic functions
zbMATH Keywords: local integrability
reviewed by: M. Vuorinen
MaRDI profile type: Publication
cites work: The Dirichlet problem for p-harmonic functions on metric spaces
cites work: Boundary continuity for quasiminimizers on metric spaces
cites work: Fat sets and pointwise boundary estimates for \(p\)-harmonic functions in metric spaces
cites work: Differentiability of Lipschitz functions on metric measure spaces
cites work: Extremal length and functional completion
cites work: Q4847438
cites work: Lectures on analysis on metric spaces
cites work: Q4693180
cites work: Quasiconformal maps in metric spaces with controlled geometry
cites work: Sobolev spaces with zero boundary values on metric spaces
cites work: Nonlinear potential theory of metric spaces
cites work: Q4458801
cites work: Polar sets on metric spaces
cites work: Regularity of quasi-minimizers on metric spaces
cites work: On the definition and properties of p-superharmonic functions.
cites work: On Harnack's theorem for elliptic differential equations
cites work: Local behavior of solutions of quasi-linear equations
cites work: Harmonic functions on metric spaces
cites work: Newtonian spaces: An extension of Sobolev spaces to metric measure spaces
cites work: On harnack type inequalities and their application to quasilinear elliptic equations
description: scientific article; zbMATH DE number 2062006