Geodesic
Resistance Conditions and Applications
Analysis and Geometry in Metric Spaces, Tome 1 (2013) no. 1, pp. 276-294.

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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.
Mots-clés : Metric measure space, resistance condition, Poincaré inequality, Hausdorff content of codimension one, Hardy-Littlewood maximal function, Sobolev type inequalities 
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     author = {Kinnunen, Juha and Silvestre, Pilar},
     title = {Resistance {Conditions} and {Applications}},
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Kinnunen, Juha; Silvestre, Pilar. Resistance Conditions and Applications. Analysis and Geometry in Metric Spaces, Tome 1 (2013) no. 1, pp. 276-294. http://geodesic.mathdoc.fr/item/AGMS_2013_1_1_a12/