The parabolic Harnack inequality on non-local Dirichlet spaces in the view of pure analysis

TL;DR

This paper develops a comprehensive analytic framework for establishing parabolic Harnack inequalities on non-local Dirichlet spaces, expanding the understanding of their properties and characterizations.

Contribution

It introduces pure analytic methods to prove PHI for regular Dirichlet forms without killing parts, broadening the theoretical foundation.

Findings

Proved PHI using Nash and Moser methods.

Established equivalences with weak Harnack inequalities.

Enhanced understanding of properties contained in PHI.

Abstract

This paper provides the general theory on parabolic Harnack inequalities (PHI, for short) for regular Dirichlet forms without killing part. We prove PHI by pure analytic methods, using both Nash and Moser approaches, and yield some important properties contained in PHI. Combining our recent result on weak Harnack inequalities, we greatly enlarge the list of equivalent characterizations of PHI.