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 ( P p ( X , d ) , W p , d , m ) generated by a finite and positive Borel measure m on the ( p , d ) -Wasserstein space ( P p ( X , d ) , W p , d ) on a complete and separable metric space ( X , d ) is dense in energy. As an application, we prove that, in case the underlying metric space is a separable Banach space B , then the Wasserstein Sobolev space is reflexive (resp. uniformly convex) if B is reflexive (resp. if the dual of B is uniformly convex). Finally, we also provide sufficient conditions for the validity of Clarkson's type inequalities in the Wasserstein Sobolev space.