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# Heat Kernels and Green Functions on Metric Measure Spaces

Published:2013-02-06
Printed: Jun 2014
• Alexander Grigor'yan,
Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld, Germany
• Jiaxin Hu,
Department of Mathematical Sciences, and Mathematical Sciences Center, Tsinghua University, Beijing 100084, China
 Format: LaTeX MathJax PDF

## Abstract

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 propety, 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: Dirichlet form, heat kernel, Green function, capacity
 MSC Classifications: 35K08 - Heat kernel 28A80 - Fractals [See also 37Fxx] 31B05 - Harmonic, subharmonic, superharmonic functions 35J08 - Green's functions 46E35 - Sobolev spaces and other spaces of smooth'' functions, embedding theorems, trace theorems 47D07 - Markov semigroups and applications to diffusion processes {For Markov processes, see 60Jxx}

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