Duality properties of metric Sobolev spaces and capacity

Abstract

We study the properties of the dual Sobolev space $H^{-1, q}(\mathbb{X}) = \big(H^{1, p}(\mathbb{X})\big)'$ on a complete extended metric-topological measure space $\mathbb{X} = (X, \tau, \rm{d}, \rm{m})$ for $p\in (1, \infty)$. We will show that a crucial role is played by the strong closure $H_{{\rm{pd}}}^{ - 1, q}\left({\mathbb{X}} \right)$ of $L^q(X, \rm{m})$ in the dual $H^{-1, q}(\mathbb{X})$, which can be identified with the predual of $H^{1, p}(\mathbb{X})$. We will show that positive functionals in $H^{-1, q}(\mathbb{X})$ 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_<i>p</i>-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_<i>p</i> admits a natural dual representation in terms of such a class of Radon measures.