Harnack inequality for stable processes on $d$-sets

Volume 158 / 2003

Krzysztof Bogdan, Andrzej Stós, Paweł Sztonyk Studia Mathematica 158 (2003), 163-198 MSC: Primary 60J45; Secondary 60J35. DOI: 10.4064/sm158-2-5

Abstract

We investigate properties of functions which are harmonic with respect to $\alpha $-stable processes on $d$-sets such as the Sierpiński gasket or carpet. We prove the Harnack inequality for such functions. For every process we estimate its transition density and harmonic measure of the ball. We prove continuity of the density of the harmonic measure. We also give some results on the decay rate of harmonic functions on regular subsets of the $d$-set. In the case of the Sierpiński gasket we even obtain the Boundary Harnack Principle.

Authors

  • Krzysztof BogdanInstitute of Mathematics
    Wrocław University of Technology
    Wybrzeże Wyspiańskiego 27
    50-370 Wrocław, Poland
    e-mail
  • Andrzej StósInstitute of Mathematics
    Wrocław University of Technology
    Wybrzeże Wyspiańskiego 27
    50-370 Wrocław, Poland
    e-mail
  • Paweł SztonykInstitute of Mathematics
    Wrocław University of Technology
    Wybrzeże Wyspiańskiego 27
    50-370 Wrocław, Poland
    e-mail

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