CONSTRUCTION OF p-ENERGY AND ASSOCIATED ENERGY MEASURES ON SIERPINSKI CARPETS

We establish the existence of a scaling limit epsilon(p) of discrete p energies on the graphs approximating a generalized Sierpinski carpet for p > d(ARC), where d(ARC) is the Ahlfors regular conformal dimension of the underlying generalized Sierpinski carpet. Furthermore, the function space 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, (epsilon(2), F-2) recovers the canonical regular Dirichlet form constructed by Barlow and Bass [Ann. Inst. H. Poincare ' Probab. Statist. 25 (1989), pp. 225-257] or Kusuoka and Zhou [Probab. Theory Related Fields 93 (1992), pp. 169-196]. We also provide epsilon(p)-energy measures associated with the constructed p-energy and investigate its basic properties like self-similarity and chain rule.