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Harnack inequalities for non-local operators of variable order

Authors: Richard F. Bass and Moritz Kassmann
Journal: Trans. Amer. Math. Soc. 357 (2005), 837-850
MSC (2000): Primary 45K05; Secondary 60H10
Published electronically: July 22, 2004
MathSciNet review: 2095633
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Abstract | References | Similar Articles | Additional Information

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.

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Additional Information

Richard F. Bass
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009

Moritz Kassmann
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009 – and Institut für Angewandte Mathematik, Universität Bonn, Beringstrasse 6, D-53115 Bonn, Germany

Keywords: Harnack inequality, non-local operator, stable processes, L\'evy processes, jump processes, integral operators
Received by editor(s): May 27, 2003
Received by editor(s) in revised form: October 27, 2003
Published electronically: July 22, 2004
Additional Notes: The first author’s research was partially supported by NSF grant DMS-9988496
Article copyright: © Copyright 2004 American Mathematical Society