Probability Theory and Related Fields

, Volume 140, Issue 3–4, pp 345–381

Estimates and structure of α-harmonic functions

  • Krzysztof Bogdan
  • Tadeusz Kulczycki
  • Mateusz Kwaśnicki
Article
  • 198 Downloads

Abstract

We prove a uniform boundary Harnack inequality for nonnegative harmonic functions of the fractional Laplacian on arbitrary open set D. This yields a unique representation of such functions as integrals against measures on Dc∪ {∞} satisfying an integrability condition. The corresponding Martin boundary of D is a subset of the Euclidean boundary determined by an integral test.

Keywords

Boundary Harnack inequality Martin representation Stable process 

Mathematics Subject Classification (2000)ss

Primary: 31C35 60J50 Secondary: 31B05 60G51 

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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  • Krzysztof Bogdan
    • 1
    • 2
  • Tadeusz Kulczycki
    • 2
  • Mateusz Kwaśnicki
    • 2
  1. 1.Department of StatisticsPurdue UniversityWest LafayetteUSA
  2. 2.Institute of Mathematics and Computer ScienceWrocław University of TechnologyWrocławPoland

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