Conductive Homogeneity of Compact Metric Spaces and Construction of pp-Energy

  • Jun Kigami

    Kyoto University, Japan
Conductive Homogeneity of Compact Metric Spaces and Construction of 𝑝-Energy cover
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In the ordinary theory of Sobolev spaces on domains of Rn\mathbb{R}^n, the pp-energy is defined as the integral of fp|\nabla{f}|^p. In this paper, we try to construct a pp-energy on compact metric spaces as a scaling limit of discrete pp-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 pp-energy if pp is greater than the Ahlfors regular conformal dimension of the space. In particular, if p=2p = 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 Sierpiński crosses, where no diffusions were constructed before.