Conductive Homogeneity of Compact Metric Spaces and Construction of p-Energy.

In the ordinary theory of Sobolev spaces on domains of ℝⁿ, the p -energy is defined as the integral of |∇ f |^p. In this paper, we try to construct a p-energy on compact metric spaces as a scaling limit of discrete p-energies on a series of graphs approximating the original space. In conclusion, we propose a notion called conductive homogeneity under which one can construct a reasonable p-energy if p is greater than the Ahlfors regular conformal dimension of the space. In particular, if p = 2, then we construct a local regular Dirichlet form and show that the heat kernel associated with the Dirichlet form satisfies upper and lower sub-Gaussian type heat kernel estimates. As examples of conductively homogeneous spaces, we present new classes of square-based self-similar sets and rationally ramified Sierpmski crosses, where no diffusions were constructed before. [ABSTRACT FROM AUTHOR]