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Estimates for capacities of nodal sets and polarity criteria in recurrent Dirichlet spaces

Estimates for capacities of nodal sets and polarity criteria in recurrent Dirichlet spaces Abstract. In this work we consider an irreducible and recurrent symmetric Dirichlet space EY F on a locally compact separable metric space X with reference measure m, such that the transition function pt xY Á of the associated Hunt's process is absolutely continuous w.r.t. m, for quasi every x e X . We then give estimates for (1-order) capacities CapNv of the nodal  set Nv X fx e X X vx 0g of a function v in the extended space F , by proving Poincaree Wirtinger-type inequalities where the ``constant'' involved actually depends on CapNv . We then derive estimates for the capacity of a closed set F, similar to those proved by K.-Th.  Sturm. For example, when a Poincare-Wirtinger's inequality holds true these estimates are of the following form: 2 CapF constXvF RF À vF RÀ2 R r dr 0 vF r 3À1  where vF r mfx e X X 0 ` rxY F ` rg is the volume growth function of the Caratheodory metric r associated with the strongly local part of the Dirichlet space and RF X supx e X rxY F . From these bounds criteria for polarity of not necessarily http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Estimates for capacities of nodal sets and polarity criteria in recurrent Dirichlet spaces

Cipriani, Fabio
Forum Mathematicum , Volume 12 (1) – Dec 8, 1999
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References (16)

Publisher
de Gruyter
Copyright
Copyright © 1999 by Walter de Gruyter GmbH & Co. KG
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/form.1999.026
Publisher site
See Article on Publisher Site

Abstract

Abstract. In this work we consider an irreducible and recurrent symmetric Dirichlet space EY F on a locally compact separable metric space X with reference measure m, such that the transition function pt xY Á of the associated Hunt's process is absolutely continuous w.r.t. m, for quasi every x e X . We then give estimates for (1-order) capacities CapNv of the nodal  set Nv X fx e X X vx 0g of a function v in the extended space F , by proving Poincaree Wirtinger-type inequalities where the ``constant'' involved actually depends on CapNv . We then derive estimates for the capacity of a closed set F, similar to those proved by K.-Th.  Sturm. For example, when a Poincare-Wirtinger's inequality holds true these estimates are of the following form: 2 CapF constXvF RF À vF RÀ2 R r dr 0 vF r 3À1  where vF r mfx e X X 0 ` rxY F ` rg is the volume growth function of the Caratheodory metric r associated with the strongly local part of the Dirichlet space and RF X supx e X rxY F . From these bounds criteria for polarity of not necessarily

Journal

Forum Mathematicum – de Gruyter

Published: Dec 8, 1999

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