Laplace Operators Related to Self-Similar Measures on \u3cem\u3eR\u3c/em\u3e\u3csup\u3ed\u3c/sup\u3e

Given a bounded open subset Ω of Rd (d⩾1) and a positive finite Borel measure μ supported on ¯Ω with μ(Ω)\u3e0, 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 dim̲∞(μ). We give a sufficient condition for which the Sobolev space H1/0(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 Δμ. We also give a sufficient condition under which the Green\u27s operator associated with μ exists, and is the inverse of −Δμ. In both cases, the condition dim̲∞(μ)\u3ed−2 plays a crucial rôle. By making use of the multifractal Lq-spectrum of the measure, we investigate the condition dim̲∞(μ)\u3ed−2 for self-similar measures defined by iterated function systems satisfying or not satisfying the open set condition