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

Authors

Jiaxin Hu, Tsinghua UniversityFollow
Ka-Sing Lau, The Chinese University of Hong KongFollow
Sze-Man Ngai, Georgia Southern UniversityFollow

Document Type

Article

Publication Date

10-15-2006

Publication Title

Journal of Functional Analysis

DOI

10.1016/j.jfa.2006.07.005

ISSN

0022-1236

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 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 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 dim̲_∞(μ)>d−2 plays a crucial rôle. By making use of the multifractal L^q-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.

Recommended Citation

Hu, Jiaxin, Ka-Sing Lau, Sze-Man Ngai. 2006. "Laplace Operators Related to Self-Similar Measures on R^d." Journal of Functional Analysis, 239 (2): 542-565. doi: 10.1016/j.jfa.2006.07.005
https://digitalcommons.georgiasouthern.edu/math-sci-facpubs/609