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, Capp-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 Capp admits a natural dual representation in terms of such a class of Radon measures.

Keywords

• metric sobolev spaces
• capacity
• modulus of a family of rectifiable curves
• dynamic transport plans
• dual cheeger energy
• capacitary measures