Boundary Harnack Inequality for Α-harmonic Functions on the Sierpiński Triangle

The analysis and probability theory on fractals underwent rapid development in last twenty years, see [1, 11, 27, 28] and the references therein. Diffusion processes were constructed for the Sierpiński triangle [4, 15, 21] and more generally for some simple nested fractals [17] and Sierpiński carpets [2, 18, 20, 22]. In [25] Stós introduced a class of subordinate processes on d-sets, called α-stable processes on d-sets by analogy to the classical setting (see also [19]). Their nice scaling properties are similar to those of diffusion processes on d-sets, but their paths are no longer continuous. For the formal definition, see the Preliminaries section; here we only remark that in order to make the notion of α-stability consistent with the scaling properties mentioned above, we depart from the notation of [25]. Namely, the α-stable process below refers to the ( 2α dw )-stable process in the sense of [25]. In particular, subordination yields α ∈ (0, dw) rather than α ∈ (0, 2) as in [25]. The theory of α-stable processes on d-sets was further developed in [9, 10, 19]. In particular, it is known that the Harnack inequality holds true for nonnegative functions harmonic with respect to the α-stable process (α-harmonic functions) on a d-set F whenever there is a diffusion process on F and α ∈ (0, 1) ∪ (d, dw) [9, Theorem 7.1]. Also, it is proved in [9, Theorem 8.6] that for the Sierpiński triangle a version of the boundary Harnack inequality holds for α ∈ (0, 1) ∪ (d, dw) if domain of harmonicity is a union of fundamental cells. The main result of this article extends this result for α ∈ (0, 1) to arbitrary open sets.