Laplace operators related to self-similar measures on Rd

Given 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.