Laplace operators related to self-similar measures on [formula omitted

Abstract

Given a bounded open subset Ω of R d ( d ⩾ 1 ) and a positive finite Borel measure μ supported on Ω ¯ with μ ( Ω ) > 0 , we study a Laplace-type operator Δ 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 0 1 ( Ω ) is compactly embedded in L 2 ( Ω , μ ) , which leads to the existence of an orthonormal basis of L 2 ( 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 dim ̲ ∞ ( μ ) > d − 2 plays a crucial rôle. By making use of the multifractal L q -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.