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 [Formula: see text] generated by a finite and positive Borel measure [Formula: see text] on the [Formula: see text] -Wasserstein space [Formula: see text] on a complete and separable metric space [Formula: see text] is dense in energy. As an application, we prove that, in case the underlying metric space is a separable Banach space [Formula: see text] , then the Wasserstein Sobolev space is reflexive (resp. uniformly convex) if [Formula: see text] is reflexive (resp. if the dual of [Formula: see text] is uniformly convex). Finally, we also provide sufficient conditions for the validity of Clarkson’s type inequalities in the Wasserstein Sobolev space.