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

Abstract

AbstractWe show that the algebra of cylinder functions in the Wasserstein Sobolev space$$H^{1,q}(\mathcal {P}_p(X,\textsf{d}), W_{p, \textsf{d}}, \mathfrak {m})$$H1,q(Pp(X,d),Wp,d,m)generated by a finite and positive Borel measure$$\mathfrak {m}$$mon the$$(p,\textsf{d})$$(p,d)-Wasserstein space$$(\mathcal {P}_p(X, \textsf{d}), W_{p, \textsf{d}})$$(Pp(X,d),Wp,d)on a complete and separable metric space$$(X,\textsf{d})$$(X,d)is dense in energy. As an application, we prove that, in case the underlying metric space is a separable Banach space$$\mathbb {B}$$B, then the Wasserstein Sobolev space is reflexive (resp. uniformly convex) if$$\mathbb {B}$$Bis reflexive (resp. if the dual of$$\mathbb {B}$$Bis uniformly convex). Finally, we also provide sufficient conditions for the validity of Clarkson’s type inequalities in the Wasserstein Sobolev space.