Potential Analysis

, Volume 41, Issue 1, pp 1–29 | Cite as

On Harnack Inequality and Hölder Regularity for Isotropic Unimodal Lévy Processes

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Abstract

We prove the scale invariant Harnack inequality and regularity properties for harmonic functions with respect to an isotropic unimodal Lévy process with the characteristic exponent ψ satisfying some scaling condition. We derive sharp estimates of the potential measure and capacity of balls, and further, under the assumption that ψ satisfies the lower scaling condition, sharp estimates of the potential kernel of the underlying process. This allows us to establish the Krylov–Safonov type estimate, which is the key ingredient in the approach of Bass and Levin, that we follow.

Keywords

Lévy process Green function Harnack inequality Harmonic function Potential measure Capacity Subordinate Brownian motion 

Mathematics Subject Classifications (2010)

60J45 60G51 31B15 
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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Institute of Mathematics and Computer SciencesWrocław University of TechnologyWroclawPoland

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