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Heat Kernels and Green Functions on Metric Measure Spaces

Published online by Cambridge University Press:  20 November 2018

Alexander Grigor'yan  and
Jiaxin Hu
Alexander Grigor'yan
Affiliation:
Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany. e-mail: grigor@math.uni-bielefeld.de
Jiaxin Hu
Affiliation:
Department of Mathematical Sciences and Mathematical Sciences Center, Tsinghua University, Beijing 100084, China. e-mail: hujiaxin@mail.tsinghua.edu.cn
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Abstract

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We prove that, in a setting of local Dirichlet forms on metric measure spaces, a two-sided sub-Gaussian estimate of the heat kernel is equivalent to the conjunction of the volume doubling property, the elliptic Harnack inequality, and a certain estimate of the capacity between concentric balls. The main technical tool is the equivalence between the capacity estimate and the estimate of a mean exit time in a ball that uses two-sided estimates of a Green function in a ball.

Keywords

35K0828A8031B0535J0846E3547D07Dirichlet formheat kernelGreen functioncapacity
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 66 , Issue 3 , 01 June 2014 , pp. 641 - 699
Copyright
Copyright © Canadian Mathematical Society 2014

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