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Heat Kernels and Green Functions on Metric Measure Spaces |
55 pages
Published:2013-02-06
- Alexander Grigor'yan,
- Jiaxin Hu,
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} |