Probability Theory and Related Fields

, Volume 140, Issue 3, pp 345–381

Estimates and structure of α-harmonic functions

Authors

    • Department of StatisticsPurdue University
    • Institute of Mathematics and Computer ScienceWrocław University of Technology
  • Tadeusz Kulczycki
    • Institute of Mathematics and Computer ScienceWrocław University of Technology
  • Mateusz Kwaśnicki
    • Institute of Mathematics and Computer ScienceWrocław University of Technology
Article

DOI: 10.1007/s00440-007-0067-0

Cite this article as:
Bogdan, K., Kulczycki, T. & Kwaśnicki, M. Probab. Theory Relat. Fields (2008) 140: 345. doi:10.1007/s00440-007-0067-0

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 inequalityMartin representationStable process

Mathematics Subject Classification (2000)ss

Primary: 31C3560J50Secondary: 31B0560G51
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© Springer-Verlag 2007