Intrinsic H\"older continuity of harmonic functions

TL;DR

This paper establishes simple criteria for the intrinsic H"older continuity of harmonic functions in a general setting, showing that weak scaling properties combined with Harnack inequalities suffice for regularity.

Contribution

It provides a unified framework for proving H"older continuity of harmonic functions based on exit measures and scaling properties, extending recent results and covering cases with Green functions.

Findings

Weak scaling property implies H"older continuity via Harnack inequalities

Framework applies to settings with Green functions and intrinsic metrics

Simplifies criteria for regularity of harmonic functions in metric spaces

Abstract

In a setting, where only exit measures are given, as they are associated with a right continuous strong Markov process on a separable metric space, we provide simple criteria for scaling invariant H\"older continuity of bounded harmonic functions with respect to a distance function which, in applications, may be adapted to the special situation. In particular, already a very weak scaling property ensures that Harnack inequalities imply H\"older continuity. Our approach covers recent results by M. Kassmann and A. Mimica as well as cases, where a Green function leads to an intrinsic metric.