Weak parabolic Harnack inequality and H\"older regularity for non-local Dirichlet forms

TL;DR

This paper establishes equivalent conditions for the weak parabolic Harnack inequality in regular Dirichlet forms, leading to H"older continuity of harmonic functions without requiring upper jump smoothness.

Contribution

It generalizes existing theory by removing the upper jump smoothness condition and connects heat kernel estimates with Harnack inequalities and regularity results.

Findings

Equivalent conditions for weak parabolic Harnack inequality

H"older continuity of caloric and harmonic functions

Generalization of previous results by Chen, Kumagai, and Wang

Abstract

In this paper we give equivalent conditions for the weak parabolic Harnack inequality for general regular Dirichlet forms without killing part, in terms of local heat kernel estimates or growth lemmas. With a tail estimate on the jump measure, we obtain from these conditions the H\"older continuity of caloric and harmonic functions. Our results generalize the theory of Chen, Kumagai and Wang, in the sense that the upper jumping smoothness condition is canceled. We also derive the complete forms of Harnack inequalities from the globally non-negative versions, and obtain continuity of caloric functions with worse tails.