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