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Local and semilocal Poincare inequalities on metric spaces
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.ORCID iD: 0000-0002-9677-8321
Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, Faculty of Science & Engineering.ORCID iD: 0000-0002-1238-6751
2018 (English)In: Journal des Mathématiques Pures et Appliquées, ISSN 0021-7824, E-ISSN 1776-3371, Vol. 119, p. 158-192Article in journal (Refereed) Published
Abstract [en]

We consider several local versions of the doubling condition and Poincare inequalities on metric measure spaces. Our first result is that in proper connected spaces, the weakest local assumptions self-improve to semilocal ones, i.e. holding within every ball. We then study various geometrical and analytical consequences of such local assumptions, such as local quasiconvexity, self-improvement of Poincare inequalities, existence of Lebesgue points, density of Lipschitz functions and quasicontinuity of Sobolev functions. It turns out that local versions of these properties hold under local assumptions, even though they are not always straightforward. We also conclude that many qualitative, as well as quantitative, properties of p-harmonic functions on metric spaces can be proved in various forms under such local assumptions, with the main exception being the Liouville theorem, which fails without global assumptions. (C) 2018 Elsevier Masson SAS. All rights reserved.

Place, publisher, year, edition, pages
ELSEVIER SCIENCE BV , 2018. Vol. 119, p. 158-192
Keywords [en]
Density of Lipschitz functions; Lebesgue point; Local doubling; Metric measure space; Poincare inequality; Quasicontinuity
National Category
Mathematical Analysis
Identifiers
URN: urn:nbn:se:liu:diva-152595DOI: 10.1016/j.matpur.2018.05.005ISI: 000448496600005OAI: oai:DiVA.org:liu-152595DiVA, id: diva2:1262135
Available from: 2018-11-09 Created: 2018-11-09 Last updated: 2018-11-09

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Björn, AndersBjörn, Jana
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CiteExportLink to record
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  • apa
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  • Other style
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  • de-DE
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