Stability of parabolic Harnack inequalities

Abstract

Abstract. Let (G, E) be a graph with weights {axy} for which a parabolic Harnack inequality holds with space-time scaling exponent β ≥ 2. Suppose {a xy } is another set of weights that are comparable to {axy}. We prove that this parabolic Harnack inequality also holds for (G, E) with the weights {a xy }. We also give stable necessary and sufficient conditions for this parabolic Harnack inequality to hold.

The Dirichlet Heat Kernel in Inner Uniform Domains in Fractal-Type Spaces

“…Under the weaker assumptions of Theorem 1.12, the resistance estimate RES( ) can still be proved as in [9,Lemma 5.1]. The Feller property is already proved in Proposition 1.11 (replacing (E, F) by (E, F(Y )), we may assume that A2 and CSA( ) hold globally).…”

“…We will apply (9) to the right side of the above estimate. We also use the Leibniz rule for the energy measure, strong locality and the fact that g D is constant on each connected component of the support of φ, and the Cauchy-Schwarz inequality.…”

A stability theorem for elliptic Harnack inequalities

“…That the EHI for C implies the COI for C is proved in almost the identical way that it is done in [2,Sect. 4].…”

“…Let ϕ 1 be the cut-off function for B(x 0 , 2κ 2 S) given by the COI. Exactly as in the proof of [2,Prop. 5.7] we have…”

“…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.…”