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

TL;DR

This paper studies the structure of Wasserstein Sobolev spaces, proving density of cylinder functions, reflexivity, uniform convexity, and Clarkson's inequalities, with implications for analysis on metric measure spaces.

Contribution

It establishes the density of cylinder functions in Wasserstein Sobolev spaces and characterizes their geometric properties like reflexivity and convexity.

Findings

Cylinder functions are dense in energy in Wasserstein Sobolev spaces.

Reflexivity and uniform convexity of these spaces depend on the underlying Banach space properties.

Conditions for Clarkson's inequalities to hold in Wasserstein Sobolev spaces are provided.

Abstract

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.