Heat kernels on metric measure spaces and an application to semilinear elliptic equations

We consider a metric measure space (M, d, mu) and a heat kernel p(t) (x, y) on M satisfying certain upper and lower estimates, which depend on two parameters alpha and beta. We show that under additional mild assumptions, these parameters are determined by the intrinsic properties of the space (M, d, mu). Namely, alpha is the Hausdorff dimension of this space, whereas beta, called the walk dimension, is determined via the properties of the family of Besov spaces W-sigma,W-2 on M. Moreover, the parameters alpha and beta are related by the inequalities 2 less than or equal to beta less than or equal to alpha + 1. We prove also the embedding theorems for the space W-beta/2,W-2, and use them to obtain the existence results for weak solutions to semilinear elliptic equations on M of the form -Lu + f(x, u) = g(x), where C is the generator of the semigroup associated with p(t). The framework in this paper is applicable for a large class of fractal domains, including the generalized Sierpiniski carpet in R-n.