Elliptic Harnack inequalities for symmetric non-local Dirichlet forms

TL;DR

This paper investigates elliptic Harnack inequalities for symmetric non-local Dirichlet forms on metric measure spaces, establishing stability, regularity implications, and new equivalences with parabolic inequalities, even with state-dependent scaling and disconnected spaces.

Contribution

It introduces new characterizations and stability results for elliptic Harnack inequalities in complex non-local settings with state-dependent scales and disconnected spaces.

Findings

Stability of elliptic Harnack inequalities under regularity conditions

Implication of Harnack inequalities for Hölder regularity of harmonic functions

New equivalences between elliptic and parabolic Harnack inequalities

Abstract

We study relations and characterizations of various elliptic Harnack inequalities for symmetric non-local Dirichlet forms on metric measure spaces. We allow the scaling function be state-dependent and the state space possibly disconnected. Stability of elliptic Harnack inequalities is established under certain regularity conditions and implication for a priori H\"older regularity of harmonic functions is explored. New equivalent statements for parabolic Harnack inequalities of non-local Dirichlet forms are obtained in terms of elliptic Harnack inequalities.