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Lipschitz continuity of Cheeger-harmonic functions in metric measure spaces

Kai RajalaNageswari ShanmugalingamPekka Koskela

subject

Pure mathematicsMathematical analysisLipschitz continuityModulus of continuityCheeger-harmonicConvex metric spaceUniform continuityMetric spaceLipschitz domainPoincaré inequalityheat kerneldoubling measureMetric mapLipschitz regularitylogarithmic Sobolev inequalityMetric differentialhypercontractivityAnalysisNewtonian spaceMathematics

description

Abstract We use the heat equation to establish the Lipschitz continuity of Cheeger-harmonic functions in certain metric spaces. The metric spaces under consideration are those that are endowed with a doubling measure supporting a (1,2)-Poincare inequality and in addition supporting a corresponding Sobolev–Poincare-type inequality for the modification of the measure obtained via the heat kernel. Examples are given to illustrate the necessity of our assumptions on these spaces. We also provide an example to show that in the general setting the best possible regularity for the Cheeger-harmonic functions is Lipschitz continuity.

yearjournalcountryeditionlanguage
2003-08-01Journal of Functional Analysis
https://doi.org/10.1016/s0022-1236(02)00090-3