Eigenvalue estimates for Laplacians on measure spaces
We obtain various lower and upper estimates for the first eigenvalue of Dirichlet Laplacians defined by positive Borel measures on bounded open subsets of Rn. These Laplacians and the corresponding eigenvalue estimates differ from classical ones in that the defining measures can be singular. The Laplacians are also different from those in Kigami's theory in that the defining iterated function systems need not be post-critically finite. By using properties of self-similar measures, such as Strichartz's second-order self-similar identities, we improve some of the eigenvalue estimates.