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

Abstract

We establish the existence of a scaling limit E p \mathcal {E}_p of discrete p p -energies on the graphs approximating a generalized Sierpiński carpet for p > d A R C p > d_{\mathrm {ARC}} , where d A R C d_{\mathrm {ARC}} is the Ahlfors regular conformal dimension of the underlying generalized Sierpiński carpet. Furthermore, the function space F p \mathcal {F}_{p} defined as the collection of functions with finite p 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, ( E 2 , F 2 ) (\mathcal {E}_2, \mathcal {F}_2) recovers the canonical regular Dirichlet form constructed by Barlow and Bass [Ann. Inst. H. Poincaré Probab. Statist. 25 (1989), pp. 225–257] or Kusuoka and Zhou [Probab. Theory Related Fields 93 (1992), pp. 169–196]. We also provide E p \mathcal {E}_{p} -energy measures associated with the constructed p p -energy and investigate its basic properties like self-similarity and chain rule.