Laplace operators related to self - similar measures on R d

Given a bounded open subset Ω of Rd (d 1) and a positive finite Borel measure μ supported on Ω with μ(Ω) > 0, we study a Laplace-type operator μ that extends the classical Laplacian. We show that the properties of this operator depend on the multifractal structure of the measure, especially on its lower L∞-dimension dim∞(μ). We give a sufficient condition for which the Sobolev space H 1 0 (Ω) is compactly embedded in L2(Ω,μ), which leads to the existence of an orthonormal basis of L2(Ω,μ) consisting of eigenfunctions of μ. We also give a sufficient condition under which the Green’s operator associated with μ exists, and is the inverse of − μ. In both cases, the condition dim∞(μ) > d − 2 plays a crucial rôle. By making use of the multifractal Lq -spectrum of the measure, we investigate the condition dim∞(μ) > d − 2 for self-similar measures defined by iterated function systems satisfying or not satisfying the open set condition. © 2006 Elsevier Inc. All rights reserved.