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Analysis and Geometry in Metric Spaces

Ed. by Ritoré, Manuel

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Resistance Conditions and Applications

Juha Kinnunen / Pilar Silvestre
Published Online: 2013-10-25 | DOI: https://doi.org/10.2478/agms-2013-0007


This paper studies analytic aspects of so-called resistance conditions on metric measure spaces with a doubling measure. These conditions are weaker than the usually assumed Poincaré inequality, but however, they are sufficiently strong to imply several useful results in analysis on metric measure spaces. We show that under a perimeter resistance condition, the capacity of order one and the Hausdorff content of codimension one are comparable. Moreover, we have connections to the Sobolev inequality for compactly supported Lipschitz functions on balls as well as capacitary strong type estimates for the Hardy-Littlewood maximal function. We also consider extensions to Sobolev type inequalities with two different measures and Lorentz type estimates.

Keywords: Metric measure space; resistance condition; Poincaré inequality; Hausdorff content of codimension one; Hardy-Littlewood maximal function; Sobolev type inequalities

MSC: 46E35; 31C45

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About the article

Received: 2013-03-28

Accepted: 2013-10-04

Published Online: 2013-10-25

Citation Information: Analysis and Geometry in Metric Spaces, Volume 1, Pages 276–294, ISSN (Online) 2299-3274, DOI: https://doi.org/10.2478/agms-2013-0007.

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©2013 Versita Sp. z o.o.. This content is open access.

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