Heat kernels on metric spaces and a characterisation of constant functions
type: Item
instance of: scholarly article
title: Heat kernels on metric spaces and a characterisation of constant functions
zbMATH Open document ID: 1062.60076
author: Katarzyna Pietruska-Pałuba
published in: Manuscripta Mathematica
publication date: 2005-02-02T00:00:00Z
review text: The main result states that the only \(u\in L^ {2}(X, m)\) for which \[ \int _ {X} \int _ {X} \frac{(u(x)-u(y))^ {2}}{\rho (x, y)^ {d+d_ {w}}} \,dm(x) \,dm(y) <+\infty \] are the constants. Here \((X, \rho)\) is a locally compact metric space and \(m\) is a Borel measure on \(X\), for which the volume of the balls of radius \(r\) behaves as \(r^ {d}\). The proof uses the Dirichlet forms theory, hence the existence of a strongly continuous semigroup of contractions on \(L^ {2}(X, m)\), whose transition density satisfies some two-sided Gaussian estimate, is assumed. Examples are included, showing the relation with recent results of J. Bourgain, H. Brézis, Y. Li, P. Mironescu, L. Nirenberg; see \textit{H. Brézis} [Russ. Math. Surv. 57, No. 4, 693--708 (2002); translation from Usp. Mat. Nauk 57, No. 4, 59--74 (2002; Zbl 1072.46020)] for a detailed account of these results.
reviewed by: Liliana Popa
Mathematics Subject Classification ID: 60J35
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 2132056
zbMATH Keywords: heat kernel
zbMATH Keywords: Dirichlet form
zbMATH Keywords: \(d\)-regular measure
MaRDI profile type: Publication
full work available at URL: s00229-004-0505-6
OpenAlex ID: W2167636282
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DOI: 10.1007/S00229-004-0505-6
description: scientific article; zbMATH DE number 2132056