H\"older continuity of harmonic functions for Hunt processes with Green function

TL;DR

This paper proves that bounded harmonic functions associated with Hunt processes having a Green function are H"older continuous under certain conditions, including the existence of a Green function, a metric, and capacity estimates.

Contribution

It establishes H"older continuity of harmonic functions in a general balayage space setting with Green functions, extending classical regularity results.

Findings

Bounded harmonic functions are H"older continuous.

H"older continuity holds if the constant function 1 is harmonic.

Polynomial estimates for jumps out of balls are sufficient for regularity.

Abstract

Let $(X,\mathcal W)$ be a balayage space, $1\in \mathcal W$, or - equivalently - let $\mathcal W$ be the set of excessive functions of a Hunt process on a locally compact space $X$ with countable base such that $\mathcal W$ separates points, every function in $\mathcal W$ is the supremum of its continuous minorants and there exist strictly positive continuous $u,v\in \mathcal W$ such that $u/v\to 0$ at infinity. We suppose that there is a Green function $G>0$ for $X$, a metric $\rho$ on $X$ and a decreasing function $g\colon[0,\infty)\to (0,\infty]$ having the doubling property and a mild upper decay such that $G\approx g\circ\rho$ and the capacity of balls of radius $r$ is approximately $1/g(r)$. It is shown that bounded harmonic functions are H\"older continuous, if the constant function $1$ is harmonic and jumps out of balls admit a polynomial estimate. The latter is proven if scaling invariant Harnack inequalities hold.