The weak elliptic Harnack inequality revisited

TL;DR

This paper revisits the weak elliptic Harnack inequality, deriving it from key conditions and exploring its characterizations and consequences within the framework of Dirichlet forms on measure metric spaces.

Contribution

It provides a new derivation of the weak elliptic Harnack inequality and characterizes it in various forms for Dirichlet forms, expanding understanding of its implications.

Findings

Derived the weak elliptic Harnack inequality from capacity, tail estimate, and Poincaré inequality.

Established equivalent characterizations of the inequality for general Dirichlet forms.

Explored consequences and applications of the weak elliptic Harnack inequality.

Abstract

In this paper we firstly derive the weak elliptic Harnack inequality from the generalized capacity condition, the tail estimate of jump measure and the Poincar\'{e} inequality, for any regular Dirichlet form without killing part on a measure metric space, by using the lemma of growth and the John-Nirenberg inequality. We secondly show several equivalent characterizations of the weak elliptic Harnack inequality for any (not necessarily regular) Dirichlet form. We thirdly present some consequences of the weak elliptic Harnack inequality.