Conductive homogeneity of compact metric spaces and construction of p-energy

TL;DR

This paper introduces a framework for constructing $p$-energy on compact metric spaces via graph approximations, establishing conditions like conductive homogeneity, and applies it to novel fractal examples with heat kernel estimates.

Contribution

It proposes the notion of conductive homogeneity to define $p$-energy on metric spaces and constructs associated Dirichlet forms on new fractal examples.

Findings

Constructed $p$-energy as a scaling limit of discrete energies.

Established heat kernel estimates for the constructed Dirichlet forms.

Presented new classes of fractals with diffusions previously unconstructed.

Abstract

In the ordinary theory of Sobolev spaces on domains of $R^n$, the $p$-energy is defined as the integral of $|\nabla{f}|^p$. In this paper, we try to construct $p$-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 a new class of square-based self-similar sets and rationally ramified Sierpinski cross, where no diffusion was constructed before.