Duality properties of metric Sobolev spaces and capacity

We study the properties of the dual Sobolev space H--1,H-q (X) = (H-1,H-p(X))' on a complete extended metric-topological measure space X = (X, tau, d, m) for p is an element of (1, infinity). We will show that a crucial role is played by the strong closure H-pd(-1)'(q) (X) of L-q(X, m) in the dual H--1,H-q(X), which can be identified with the predual of H-1,H-p(X). We will show that positive functionals in H--1,H-q(X) can be represented as a positive Radon measure and we will charaterize their dual norm in terms of a suitable energy functional on nonparametric dynamic plans. As a byproduct, we will show that for every Radon measure mu with finite dual Sobolev energy, Cap(p)-negligible sets are also mu-negligible and good representatives of Sobolev functions belong to L-1(X,mu). We eventually show that the Newtonian-Sobolev capacity Cap(p) admits a natural dual representation in terms of such a class of Radon measures.