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We consider a metric measure space (M;d; ) and aheat kernel pt(x;y )o nM satisfying certain upper and lower estimates, which depend on two parameters and . We show that under additional mild assump- tions, these parameters are determined by the intrinsic properties of the space (M;d; ). Namely, is the Hausdor dimension of this space, whereas , called the walk dimension, is determined via the properties of the family of Besov spaces W;2 on M. Moreover, the parameters and are related by the inequalities 2 +1 . We prove also the embedding theorems for the space W=2;2, and use them to obtain the existence results for weak solutions to semilinear elliptic equations on M of the form L u +f(x;u )= g(x); whereL is the generator of the semigroup associated with pt. The framework in this paper is applicable for a large class of fractal do- mains, including the generalized Sierpi nski carpet in Rn.

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