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.

Pages 1-35 are the original paper; pages 36-40 are corrections to the results of section 3

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