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Forum Mathematicum

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Ed. by Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna


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Volume 30, Issue 5

Issues

The Davies method revisited for heat kernel upper bounds of regular Dirichlet forms on metric measure spaces

Jiaxin Hu / Xuliang Li
Published Online: 2018-02-08 | DOI: https://doi.org/10.1515/forum-2017-0072

Abstract

We apply the Davies method to prove that for any regular Dirichlet form on a metric measure space, an off-diagonal stable-like upper bound of the heat kernel is equivalent to the conjunction of the on-diagonal upper bound, a cutoff inequality on any two concentric balls, and the jump kernel upper bound, for any walk dimension. If in addition the jump kernel vanishes, that is, if the Dirichlet form is strongly local, we obtain a sub-Gaussian upper bound. This gives a unified approach to obtaining heat kernel upper bounds for both the non-local and the local Dirichlet forms.

Keywords: Heat kernel; Dirichlet form; cutoff inequality on balls; Davies method

MSC 2010: 35K08; 28A80; 60J35

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About the article


Received: 2017-04-04

Revised: 2018-01-14

Published Online: 2018-02-08

Published in Print: 2018-09-01


Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11371217

Funding Source: Ministry of Education of the People’s Republic of China

Award identifier / Grant number: 20130002110003

The first author was supported by NSFC no. 11371217, SRFDP no. 20130002110003.


Citation Information: Forum Mathematicum, Volume 30, Issue 5, Pages 1129–1155, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2017-0072.

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