Construction of $p$-energy and associated energy measures on Sierpiński carpets

Abstract

We establish the existence of a scaling limit ${\mathcal{E}}_p$ of discrete $p$-energies on the graphs approximating generalized Sierpiński carpets for $p>{\dim }_{\text{ARC}}(\text{SC})$, where ${\dim }_{\text{ARC}}(\text{SC})$ is the Ahlfors regular conformal dimension of the underlying generalized Sierpiński 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.

80 pages, 5 figures; the title has been changed, upgraded results including generalized Sierpinski carpets and acknowledgments added

References in corpus (1)

• Sobolev spaces and Poincaré inequalities on the Vicsek fractal

Cited by in corpus (3)

• Heat kernel-based p-energy norms on metric measure spaces
• Construction of $p$-energy measures associated with strongly local $p$-energy forms
• Preserving Besov (fractional Sobolev) energies under sphericalization and flattening