Spectral theory for the fractal Laplacian in the context of h-sets

Abstract

An h-set is a nonempty compact subset of the Euclidean n-space which supports a finite Radon measure for which the measure of balls centered on the subset is essentially given by the image of their radius by a suitable function h. In most cases of interest such a subset has Lebesgue measure zero and has a fractal structure. Let Ω be a bounded C∞ domain in with Γ ⊂ Ω. Let where (−Δ)−1 is the inverse of the Dirichlet Laplacian in Ω and trΓ is, say, trace type operator. The operator B, acting in convenient function spaces in Ω, is studied. Estimations for the eigenvalues of B are presented, and generally shown to be dependent on h, and the smoothness of the associated eigenfunctions is discussed. Some results on Besov spaces of generalised smoothness on and on domains which were obtained in the course of this work are also presented, namely pointwise multipliers, the existence of a universal extension operator, interpolation with function parameter and mapping properties of the Dirichlet Laplacian. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim