Let 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 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.
Martin T. BARLOW. Richard F. BASS. Takashi KUMAGAI. "Stability of parabolic Harnack inequalities on metric measure spaces." J. Math. Soc. Japan 58 (2) 485 - 519, April, 2006. https://doi.org/10.2969/jmsj/1149166785