Conductive Homogeneity of Compact Metric Spaces and Construction of 𝑝-Energy
Jun Kigami
Abstract
In the ordinary theory of Sobolev spaces on domains of R n , the p-energy is defined as the integral of jrf j 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 D 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 squarebased self-similar sets and rationally ramified Sierpiński crosses, where no diffusions were constructed before.In memory of the late Professors Robert S. Strichartz and Ka-Sing Lau, who were my dearest friends and daring explorers of the frontiers