Harnack inequality for stable processes on d-sets

Abstract

We investigate properties of functions which are harmonic with respect to -stable processes on d-sets such as the Sierpi«ski 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 Sierpi«ski gasket we even obtain the Boundary Harnack Principle.