(1,p)$(1,p)$‐Sobolev spaces based on strongly local Dirichlet forms

Abstract

In the framework of quasi‐regular strongly local Dirichlet form on admitting minimal ‐dominant measure , we construct a natural ‐energy functional on and ‐Sobolev space for . In this paper, we establish the Clarkson‐type inequality for . As a consequence, is a uniformly convex Banach space, hence it is reflexive. Based on the reflexivity of , we prove that (generalized) normal contraction operates on , which has been shown in the case of various concrete settings, but has not been proved for such a general framework. Moreover, we prove that ‐capacity for open set admits an equilibrium potential with ‐a.e. and ‐a.e. on .