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

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.