Laplace Operators Related to Self-Similar Measures on R^d

Given a bounded open subset Ω of R^d(d ≥ 1) and a positive finite Borel measure μ supported on over(Ω, -) with μ (Ω) > 0, we study a Laplace-type operatorΔ_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 lowerL^∞ -dimensionunder(dim, {combining low line})_∞ (μ) . We give a sufficient condition for which the Sobolev space H₀¹ (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 Δ_μ. We also give a sufficient condition under which the Green's operator associated with μ exists, and is the inverse of - Δ_μ. In both cases, the condition under(dim, {combining low line})_∞ (μ) > d - 2 plays a crucial rôle. By making use of the multifractal L^q-spectrum of the measure, we investigate the condition under(dim, {combining low line})_∞ (μ) > d - 2 for self-similar measures defined by iterated function systems satisfying or not satisfying the open set condition.