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.
Funder
Institute for Advanced Study, Technische Universität München
Austrian Science Fund
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Analysis