# Resistance Conditions and Applications

Juha Kinnunen; Pilar Silvestre

Analysis and Geometry in Metric Spaces (2013)

- Volume: 1, page 276-294
- ISSN: 2299-3274

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topJuha Kinnunen, and Pilar Silvestre. "Resistance Conditions and Applications." Analysis and Geometry in Metric Spaces 1 (2013): 276-294. <http://eudml.org/doc/266944>.

@article{JuhaKinnunen2013,

abstract = {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.},

author = {Juha Kinnunen, Pilar Silvestre},

journal = {Analysis and Geometry in Metric Spaces},

keywords = {Metric measure space; resistance condition; Poincaré inequality; Hausdorff content of codimension one; Hardy-Littlewood maximal function; Sobolev type inequalities; metric measure space},

language = {eng},

pages = {276-294},

title = {Resistance Conditions and Applications},

url = {http://eudml.org/doc/266944},

volume = {1},

year = {2013},

}

TY - JOUR

AU - Juha Kinnunen

AU - Pilar Silvestre

TI - Resistance Conditions and Applications

JO - Analysis and Geometry in Metric Spaces

PY - 2013

VL - 1

SP - 276

EP - 294

AB - 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.

LA - eng

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

UR - http://eudml.org/doc/266944

ER -

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