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Boundary Harnack inequality for α-harmonic functions on the Sierpiński triangle

Kaleta K., Kwaśnicki M.

Probability and Mathematical Statistics

2010

Vol. 30, Fasc. 2

353--368

We prove a uniform boundary Harnack inequality for nonnegative functions harmonic with respect to α-stable process on the Sierpiński triangle, where α ∈ (0,1). Our result requires no regularity assumptions on the domain of harmonicity.

Spectral gap estimate for stable processes on arbitrary bounded open sets

Kwaśnicki M.

Probability and Mathematical Statistics

2008

Vol. 28, Fasc. 1

163--167

In the paper we prove the lower bound estimate λD2 −λDD1≥c(λD1)−d−α(diamD−d−α) for the spectral gap of the Dirichlet fractional Laplacian (−(−Δ)α/2) on an arbitrary bounded open set D⊂ Rd.