Calculus of Variations and Geometric Measure Theory

G. E. Sodini

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

created by sodini on 22 Dec 2022
modified on 25 Sep 2023


Published Paper

Inserted: 22 dec 2022
Last Updated: 25 sep 2023

Journal: Calculus of Variations and Partial Differential Equations
Volume: 62
Number: 7
Year: 2022

ArXiv: 2212.10955 PDF


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.