A new formulation of Harnackʼs inequality for nonlocal operators

Moritz Kassmann

doi:10.1016/j.crma.2011.04.014

Potential Theory/Partial Differential Equations

A new formulation of Harnackʼs inequality for nonlocal operators

Presented by: Alain Bensoussan

Moritz Kassmann ¹

¹ University of Bielefeld, Department of Mathematics, Postfach 10 01 31, 33501 Bielefeld, Germany

Comptes Rendus. Mathématique, Volume 349 (2011) no. 11-12, pp. 637-640.

Abstract
Résumé

We provide a new formulation of Harnackʼs inequality for nonlocal operators. In contrast to previous versions we do not assume harmonic functions to have a sign. The version of Harnackʼs inequality given here generalizes Harnackʼs classical result from 1887 to nonlocal situations. As a consequence we derive Hölder regularity estimates by an extension of Moserʼs method. The inequality that we propose is equivalent to Harnackʼs original formulation but seems to be new even for the Laplace operator.

Nous fournissons une nouvelle formulation de lʼinégalité de Harnack pour des opérateurs non-locaux. En contraste avec les versions précédentes, nous nʼavons pas à supposer que les fonctions harmoniques sont de signe constant. La version de lʼinégalité de Harnack donnée ici généralise le résultat classique de Harnack datant de 1887 pour les cas non-locaux. Conséquemment, on obtient des estimations de la régularité Hölder grâce à une extension de la méthode de Moser. Lʼinégalité que nous proposons est équivalente à la formulation originale de lʼinégalité de Harnack mais semble être nouvelle y compris pour lʼopérateur de Laplace.

Received: 2010-12-31
Accepted: 2011-04-21
Published online: 2011-06-01

DOI: 10.1016/j.crma.2011.04.014

Author's affiliations:

Moritz Kassmann ¹

¹ University of Bielefeld, Department of Mathematics, Postfach 10 01 31, 33501 Bielefeld, Germany

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     title = {A new formulation of {Harnack's} inequality for nonlocal operators},
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Moritz Kassmann. A new formulation of Harnackʼs inequality for nonlocal operators. Comptes Rendus. Mathématique, Volume 349 (2011) no. 11-12, pp. 637-640. doi : 10.1016/j.crma.2011.04.014. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.04.014/

References
Cited by

[1] R.F. Bass; D.A. Levin Harnack inequalities for jump processes, Potential Anal., Volume 17 (2002) no. 4, pp. 375-388

[2] R.F. Bass; D.A. Levin Transition probabilities for symmetric jump processes, Trans. Amer. Math. Soc., Volume 354 (2002) no. 7, pp. 2933-2953 (electronic)

[3] K. Bogdan; P. Sztonyk Estimates of potential kernel and Harnackʼs inequality for anisotropic fractional Laplacian, Stud. Math., Volume 181 (2007) no. 2, pp. 101-123

[4] Zh.-Qing Chen; T. Kumagai Heat kernel estimates for stable-like processes on d-sets, Stochastic Process. Appl., Volume 108 (2003) no. 1, pp. 27-62

[5] W. Hansen; J. Bliedtner Potential Theory: An Analytic and Probabilistic Approach to Balayage (Universitext), Springer, Berlin, 1986

[6] C.-G. Axel Harnack Die Grundlagen der Theorie des logarithmischen Potentiales und der eindeutigen Potentialfunktion in der Ebene, Teubner, Leipzig, 1887 (see also The Cornell Library Historical Mathematics Monographs online)

[7] M. Kassmann, The classical Harnack inequality fails for nonlocal operators, Preprint No. 360, Sonderforschungsbereich 611, Universität Bonn, 2007.

[8] T. Komatsu Uniform estimates for fundamental solutions associated with non-local Dirichlet forms, Osaka J. Math., Volume 32 (1995) no. 4, pp. 833-860

[9] N.S. Landkof Foundations of Modern Potential Theory, Springer-Verlag, New York, 1972 (translated from the Russian by A.P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band 180)

[10] J. Moser On Harnackʼs theorem for elliptic differential equations, Comm. Pure Appl. Math., Volume 14 (1961), pp. 577-591

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