Laplace Operators Related to Self-Similar Measures on Rd
Journal of Functional Analysis
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 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's operator associated with μ exists, and is the inverse of −Δμ. In both cases, the condition dim̲∞(μ)>d−2 plays a crucial rôle. By making use of the multifractal Lq-spectrum of the measure, we investigate the condition dim̲∞(μ)>d−2 for self-similar measures defined by iterated function systems satisfying or not satisfying the open set condition.
Hu, Jiaxin, Ka-Sing Lau, Sze-Man Ngai.
"Laplace Operators Related to Self-Similar Measures on Rd."
Journal of Functional Analysis, 239 (2): 542-565.
doi: 10.1016/j.jfa.2006.07.005 source: https://www.sciencedirect.com/science/article/pii/S0022123606003077?via%3Dihub