## Journal of the Mathematical Society of Japan

### Stability of parabolic Harnack inequalities on metric measure spaces

#### Abstract

Let $(X,d,\mu)$ 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 $\beta\ge 2$ 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

https://projecteuclid.org/euclid.jmsj/1149166785

Digital Object Identifier
doi:10.2969/jmsj/1149166785

Mathematical Reviews number (MathSciNet)
MR2228569

Zentralblatt MATH identifier
1102.60064

#### 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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