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\u27s theory in that the defining iterated function systems need not be post-critically finite. By using properties of self-similar measures, such as Strichartz\u27s second-order self-similar identities, we improve some of the eigenvalue estimates