Construction of $p$-energy and associated energy measures on Sierpi\'{n}ski carpets

Abstract

We establish the existence of a scaling limit $\mathcal{E}_p$ of discrete $p$-energies on the graphs approximating generalized Sierpi\'{n}ski carpets for $p > \dim_{\text{ARC}}(\textsf{SC})$, where $\dim_{\text{ARC}}(\textsf{SC})$ is the Ahlfors regular conformal dimension of the underlying generalized Sierpi\'{n}ski carpet. Furthermore, the function space $\mathcal{F}_{p}$ defined as the collection of functions with finite $p$-energies is shown to be a reflexive and separable Banach space that is dense in the set of continuous functions with respect to the supremum norm. In particular, $(\mathcal{E}_2, \mathcal{F}_2)$ recovers the canonical regular Dirichlet form constructed by Barlow and Bass or Kusuoka and Zhou. We also provide $\mathcal{E}_{p}$-energy measures associated with the constructed $p$-energy and investigate its basic properties like self-similarity and chain rule.