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

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 ${\mathbb R}^n $ with Γ ⊂ Ω. Letwhere (−Δ) −1 is the inverse of the Dirichlet Laplacian in Ω and tr Γ 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 ${{\bb R}^n} $ 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