Harnack inequality for stable processes on $d$-sets
Volume 158 / 2003
Studia Mathematica 158 (2003), 163-198 MSC: Primary 60J45; Secondary 60J35. DOI: 10.4064/sm158-2-5
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.