Laplace operators related to self-similar measures on R-d
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AbstractGiven a bounded open subset Omega of R-d (d >= 1) and a positive finite Borel measure mu supported on Omega with mu(Omega) > 0, we study a Laplace-type operator Delta(mu) 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-infinity-dimension dim(infinity)(mu). We give a sufficient condition for which the Sobolev space H-0(1)(Omega) is compactly embedded in L-2(Omega, mu), which leads to the existence of an orthonormal basis of L-2(Omega, mu) consisting of eigenfunctions of Delta(mu). We also give a sufficient condition under which the Green's operator associated with mu exists, and is the inverse of -Delta(mu). In both cases, the condition dim(infinity)(mu) > d - 2 plays a crucial role. By making use of the multifractal L-q-spectrum of the measure, we investigate the condition dim(infinity)(mu) > d - 2 for self-similar measures defined by iterated function systems satisfying or not satisfying the open set condition. (c) 2006 Elsevier Inc. All rights reserved.
All Author(s) ListHu JX, Lau KS, Ngai SM
Journal nameJournal of Functional Analysis
Volume Number239
Issue Number2
Pages542 - 565
LanguagesEnglish-United Kingdom
Keywordseigenfunction; eigenvalue; L-infinity-dimension; L-q -spectrum; Laplacian; self-similar measure; upper regularity of a measure
Web of Science Subject CategoriesMathematics; MATHEMATICS

Last updated on 2021-25-01 at 23:55