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Harnack inequalities for non-local operators of variable order

Author(s): Richard F. Bass; Moritz Kassmann
Journal: Trans. Amer. Math. Soc. 357 (2005), 837-850.
MSC (2000): Primary 45K05; Secondary 60H10
Posted: July 22, 2004
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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
Email: bass@math.uconn.edu

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
Email: kassmann@math.uconn.edu

DOI: 10.1090/S0002-9947-04-03549-4
PII: S 0002-9947(04)03549-4
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
Posted: July 22, 2004
Additional Notes: The first author's research was partially supported by NSF grant DMS-9988496
Copyright of article: Copyright 2004, American Mathematical Society

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