Stability of parabolic Harnack inequalities on metric measure spaces

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 spacetime 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.

Heat kernel bounds for the Laplacian on metric graphs of polygonal tilings

“…Of course, the estimates are not necessarily sharp and the time 1 3 β 2 appearing in (9) is hence a bit arbitrary. Nevertheless, it gives a hint how things change in dependence of the tiling.…”

A stability theorem for elliptic Harnack inequalities

“…All the key ideas are present in the infinite graph case and we avoid some unpleasant technicalities. It is straightforward to extend our results to metric measure Dirichlet spaces in a manner very similar to how [3] extended [2]; see Section 7.…”

“…See [3] for the definitions of all terms introduced in this subsection. Let (X, d, µ) be a metric measure space such that the metric is geodesic and X has infinite diameter.…”

“…If the state space is regular enough to have a large class of nice cut-off functions, then Grigor'yan [8] and Saloff-Coste [12] independently proved that the PHI holds if and only if both volume doubling and a Poincaré inequality hold. This was extended to the case where such nice cut-off functions need not exist in [2] and [3]. The latter papers allow state spaces that have fractal structure or that have large numbers of obstructions.…”

Harnack Inequalities: An Introduction

“…An elliptic Harnack inequality is used by Barlow and Bass to construct a Brownian motion on the Sierpiński carpet in [114]; see [115,116] for related results. A parabolic Harnack inequality with a non-diffusive spacetime scaling is proved on infinite connected weighted graphs [117]. Moreover it is shown that this inequality is stable under bounded transformations of the conductances.…”