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

In the framework of quasi-regular strongly local Dirichlet form (E, D(E)) on L-2(X; m) admitting minimal E-dominant measure mu, we construct a natural p-energy functional (E-p, D(E-p)) on L-p(X; m) and (1, p)-Sobolev space (H-1,H-p(X), parallel to center dot parallel to(H1,p)) for p is an element of]1, +infinity[. In this paper, we establish the Clarkson-type inequality for (H-1,H-p(X), parallel to center dot parallel to(H1,p)). As a consequence, (H-1,H-p(X), parallel to center dot parallel to(H1,p)) is a uniformly convex Banach space, hence it is reflexive. Based on the reflexivity of (H-1,H-p(X), parallel to center dot parallel to(H1,p)), we prove that (generalized) normal contraction operates on (E-p, D (E-p)), 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 (1, p)- capacity Cap(1,p) (A) < infinity for open set A admits an equilibrium potential e(A) is an element of D(E-p) with 0 = <= e(A) <= 1 m-a.e. and e(A) = 1 m.-a.e. on A.