Heat kernel estimates and parabolic Harnack inequalities for symmetric Dirichlet forms

Zhen Qing Chen, Takashi Kumagai, Jian Wang*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)

Abstract

In this paper, we consider the following symmetric Dirichlet forms on a metric measure space (M,d,μ): E(f,g)=E(c)(f,g)+∫M×M(f(x)−f(y))(g(x)−g(y))J(dx,dy), where E(c) is a strongly local symmetric bilinear form and J(dx,dy) is a symmetric Radon measure on M×M. Under general volume doubling condition on (M,d,μ) and some mild assumptions on scaling functions, we establish stability results for upper bounds of heat kernel (resp. two-sided heat kernel estimates) in terms of the jumping kernels, the cut-off Sobolev inequalities, and the Faber-Krahn inequalities (resp. the Poincaré inequalities). We also obtain characterizations of parabolic Harnack inequalities. Our results apply to symmetric diffusions with jumps even when the underlying spaces have walk dimensions larger than 2.

Original languageEnglish
Article number107269
JournalAdvances in Mathematics
Volume374
DOIs
Publication statusPublished - 2020 Nov 18
Externally publishedYes

Keywords

  • Cut-off Sobolev inequality
  • Heat kernel estimate
  • Metric measure space
  • Parabolic Harnack inequality
  • Stability
  • Symmetric Dirichlet form

ASJC Scopus subject areas

  • Mathematics(all)

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