, Volume 39, Issue 1, pp 1-11

Continuity of Harmonic Functions for Non-local Markov Generators

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Abstract

In this paper, we treat a priori estimates of harmonic functions for jump processes associated with non-local operators. Let $\mathcal{L}$ be a non-local operator given by $$\mathcal{L}u(x) = \int_{\mathbb{R}^{d} \backslash \{0\}}(u(x+h)-u(x)-h \cdot \nabla u(x) 1_{\{|h| \leq 1\}}) n(x,h)dh.$$ Under some conditions on n(x,h), we prove the Hölder continuity and the uniform continuity of $\mathcal{L}$ -harmonic functions. Our results are extensions of those obtained by Bass and Kassmann (Commun Part Diff Eq 30:1249–1259, 2005).