Hölder continuity of harmonic functions for Hunt processes with Green function

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 $g:[0,\infty )\to (0,\infty ]$ 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 Hölder 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.

arXiv admin note: text overlap with arXiv:1410.3067