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Harnack inequalities for Hunt processes with Green function

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 $ρ$ on $X$ and a decreasing function $g\colon[0,\infty)\to (0,\infty]$ having the doubling property and a mild upper decay near $0$ such that $G\approx g\circρ$ (which is equivalent to a $3G$-inequality). Then the corresponding capacity for balls of radius $r$ is bounded by a constant multiple of $1/g(r)$. Assuming that reverse inequalities hold as well and that jumps of the process, when starting at neighboring points, are related in a suitable way, it is proven that positive harmonic functions satisfy scaling invariant Harnack inequalities. Provided that the Ikeda-Watanabe formula holds, sufficient conditions for this relation are given. This shows that rather general Lévy processes are covered by this approach.