LAPLACE OPERATORS RELATED TO SELF-SIMILAR MEASURES ON Rd

Abstract. 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 op-erator 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 H10 (Ω) is compactly embedded in L 2(Ω, µ), which leads to the existence of an or-thonormal 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. 1