Intrinsic metrics and Lipschitz functions
type: Item
instance of: scholarly article
title: Intrinsic metrics and Lipschitz functions
zbMATH Open document ID: 1031.31006
author: Francis Hirsch
published in: Journal of Evolution Equations
publication date: 2004-02-22T00:00:00Z
review text: The author studies the notion of measurable metrics and Lipschitz functions w.r.t. such metrics -- originally introduced by \textit{N. Weaver} [Stud. Math. 115, 277-289 (1995; Zbl 0839.46014)] -- on Dirichlet structures and, in particular, abstract Wiener spaces. In the latter case it is shown that the intrinsic metric [in the sense of \textit{M. Biroli} and \textit{U. Mosco}, Ann. Math. Pura Appl. 169, 125-181 (1995; Zbl 0851.31008)] coincides with Weaver's measurable metric. For more general local Dirichlet structures supporting a carré du champ operator this is still an open problem. In the setting of such Dirichlet spaces the Lipschitz-continuous functions in \(L^2\) and \(L^\infty\) are characterized in terms of the carré du champ operator which, in the case of abstract Wiener spaces, leads to a new characterization of Enchev-and-Stroock's H-Lipschitz functions [\textit{O. Enchev} and \textit{D. W. Stroock}, Ann. Probab. 21, 25-33 (1993; Zbl 0773.60042)].
Mathematics Subject Classification ID: 31C25
Mathematics Subject Classification ID: 46E15
Mathematics Subject Classification ID: 46E50
Mathematics Subject Classification ID: 53C17
Mathematics Subject Classification ID: 60H99
zbMATH DE Number: 1921147
zbMATH Keywords: intrinsic metric
zbMATH Keywords: measurable metric
zbMATH Keywords: Lipschitz function
zbMATH Keywords: abstract Wiener space
zbMATH Keywords: Dirichlet form
zbMATH Keywords: carré du champ operator
reviewed by: R. L. Schilling
MaRDI profile type: Publication
full work available at URL: s000280300001
OpenAlex ID: W4247114539
DOI: 10.1007/S000280300001
description: scientific article; zbMATH DE number 1921147