Density of subalgebras of Lipschitz functions in metric Sobolev spaces and applications to Wasserstein Sobolev spaces
Massimo Fornasier, Giuseppe Savaré, Giacomo Enrico Sodini
Abstract
We prove a general criterion for the density in energy of suitable subalgebras of Lipschitz functions in the metric Sobolev space H1,p(X,d,m) associated with a positive and finite Borel measure m in a separable and complete metric space (X,d). We then provide a relevant application to the case of the algebra of cylinder functions in the Wasserstein Sobolev space H1,2(P2(M),W2,m) arising from a positive and finite Borel measure m on the Kantorovich-Rubinstein-Wasserstein space (P2(M),W2) of probability measures in a finite dimensional Euclidean space, a complete Riemannian manifold, or a separable Hilbert space M. We will show that such a Sobolev space is always Hilbertian, independently of the choice of the reference measure m so that the resulting Cheeger energy is a Dirichlet form. We will eventually provide an explicit characterization for the corresponding notion of m-Wasserstein gradient, showing useful calculus rules and its consistency with the tangent bundle and the Γ-calculus inherited from the Dirichlet form.