Duality properties of metric Sobolev spaces and capacity

We study the properties of the dual Sobolev space H-1;q(X) = - H1;p(X) '0 on a complete extended metric-topological measure space X = (X; ⊤; d;m) for p 2 (1;1). We will show that a crucial role is played by the strong closure H-1;q pd (X) of Lq(X;m) in the dual H-1;q(X), which can be identified with the predual of H1;p(X). We will show that positive functionals in H-1;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 μ with finite dual Sobolev energy, Capp-negligible sets are also μ-negligible and good representatives of Sobolev functions belong to L1(X; μ). We eventually show that the Newtonian-Sobolev capacity Capp admits a natural dual representation in terms of such a class of Radon measures.