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Harnack inequalities for non-local operators of variable order
Authors:Richard F Bass  Moritz Kassmann
Institution:Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009 ; Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009 -- and Institut für Angewandte Mathematik, Universität Bonn, Beringstrasse 6, D-53115 Bonn, Germany
Abstract:We consider harmonic functions with respect to the operator

\begin{displaymath}\mathcal{L} u(x)=\int u(x+h)-u(x)-1_{(\vert h\vert\leq 1)} h\cdot \nabla u(x)] n(x,h) \, dh. \end{displaymath}

Under suitable conditions on $n(x,h)$ we establish a Harnack inequality for functions that are nonnegative and harmonic in a domain. The operator $\mathcal{L}$ is allowed to be anisotropic and of variable order.

Keywords:Harnack inequality  non-local operator  stable processes  L\'evy processes  jump processes  integral operators
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