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
DOI:
https://doi.org/10.1090/S0002-9947-04-03549-4
Published electronically:
July 22, 2004
MathSciNet review:
2095633
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: We consider harmonic functions with respect to the operator \[ \mathcal {L} u(x)=\int [u(x+h)-u(x)-1_{(|h|\leq 1)} h\cdot \nabla u(x)] n(x,h) dh. \] 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
Keywords:
Harnack inequality,
non-local operator,
stable processes,
Lévy 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
