Stability of parabolic Harnack inequalities on metric measure spaces

Martin T. Barlow*, Richard F. Bass, Takashi Kumagai

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

70 Citations (Scopus)


Let (X, d, μ) be a metric measure space with a local regular Dirichlet form. We give necessary and sufficient conditions for a parabolic Harnack inequality with global space-time scaling exponent β ≥ 2 to hold. We show that this parabolic Harnack inequality is stable under rough isometries. As a consequence, once such a Harnack inequality is established on a metric measure space, then it holds for any uniformly elliptic operator in divergence form on a manifold naturally defined from the graph approximation of the space.

Original languageEnglish
Pages (from-to)485-519
Number of pages35
JournalJournal of the Mathematical Society of Japan
Issue number2
Publication statusPublished - 2006 Apr
Externally publishedYes


  • Anomalous diffusion
  • Green functions
  • Harnack inequality
  • Poincaré inequality
  • Rough isometry
  • Sobolev inequality
  • Volume doubling

ASJC Scopus subject areas

  • Mathematics(all)


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