Laplace operators related to self-similar measures on Rd

Given a bounded open subset Ω of Rd (d⩾1) and a positive finite Borel measure μ supported on with μ(Ω)>0, we study a Laplace-type operator Δμ 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 . We give a sufficient condition for which the Sobolev space is compactly embedded in L2(Ω,μ), which leads to the existence of an orthonormal basis of L2(Ω,μ) consisting of eigenfunctions of Δμ. We also give a sufficient condition under which the Green's operator associated with μ exists, and is the inverse of −Δμ. In both cases, the condition plays a crucial rôle. By making use of the multifractal Lq-spectrum of the measure, we investigate the condition for self-similar measures defined by iterated function systems satisfying or not satisfying the open set condition.