The general class of Wasserstein Sobolev spaces: density of cylinder functions, reflexivity, uniform convexity and Clarkson's inequalities

Autor:	Sodini, Giacomo Enrico
Rok vydání:	2022
Předmět:	Mathematics - Functional Analysis Mathematics - Metric Geometry 46E36 49Q22 46B10 46B20
Druh dokumentu:	Working Paper
Popis:	We show that the algebra of cylinder functions in the Wasserstein Sobolev space $H^{1,q}(\mathcal{P}_p(X,\mathsf{d}), W_{p, \mathsf{d}}, \mathfrak{m})$ generated by a finite and positive Borel measure $\mathfrak{m}$ on the $(p,\mathsf{d})$-Wasserstein space $(\mathcal{P}_p(X,\mathsf{d}), W_{p, \mathsf{d}})$ on a complete and separable metric space $(X,\mathsf{d})$ is dense in energy. As an application, we prove that, in case the underlying metric space is a separable Banach space $\mathbb{B}$, then the Wasserstein Sobolev space is reflexive (resp.~uniformly convex) if $\mathbb{B}$ is reflexive (resp.~if the dual of $\mathbb{B}$ is uniformly convex). Finally, we also provide sufficient conditions for the validity of Clarkson's type inequalities in the Wasserstein Sobolev space. Comment: 35 pages - final version
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/2212.10955 Zobrazit plný text záznamu View this record from Arxiv