Journal of the Mathematical Society of Japan

Stability of parabolic Harnack inequalities on metric measure spaces

Martin T. BARLOW, Richard F. BASS, and Takashi KUMAGAI

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Abstract

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.

Article information

Source
J. Math. Soc. Japan, Volume 58, Number 2 (2006), 485-519.

Dates
First available in Project Euclid: 1 June 2006

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1149166785

Digital Object Identifier
doi:10.2969/jmsj/1149166785

Mathematical Reviews number (MathSciNet)
MR2228569

Zentralblatt MATH identifier
1102.60064

Subjects
Primary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07]
Secondary: 31B05: Harmonic, subharmonic, superharmonic functions 31C25: Dirichlet spaces

Keywords
Harnack inequality volume doubling Green functions Poincaré inequality Sobolev inequality rough isometry anomalous diffusion

Citation

BARLOW, Martin T.; BASS, Richard F.; KUMAGAI, Takashi. Stability of parabolic Harnack inequalities on metric measure spaces. J. Math. Soc. Japan 58 (2006), no. 2, 485--519. doi:10.2969/jmsj/1149166785. https://projecteuclid.org/euclid.jmsj/1149166785


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