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Duality properties of metric Sobolev spaces and capacity

  • Received: 04 February 2020 Accepted: 08 July 2020 Published: 20 July 2020
  • 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.

    Citation: Luigi Ambrosio, Giuseppe Savaré. Duality properties of metric Sobolev spaces and capacity[J]. Mathematics in Engineering, 2021, 3(1): 1-31. doi: 10.3934/mine.2021001

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  • 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.


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