Density of subalgebras of Lipschitz functions in metric Sobolev spaces and applications to Wasserstein Sobolev spaces

We prove a general criterion for the density in energy of suitable subalgebras of Lipschitz functions in the metric-Sobolev space $H^{1,p}(X,\mathsf{d},\mathfrak{m})$ associated with a positive and finite Borel measure $\mathfrak{m}$ in a separable and complete metric space $(X,\mathsf{d})$. We then provide a relevant application to the case of the algebra of cylinder functions in the Wasserstein Sobolev space $H^{1,2}({\mathcal{P}}_2(\mathbb{M}),W_2,\mathfrak{m})$ arising from a positive and finite Borel measure $\mathfrak{m}$ on the Kantorovich-Rubinstein-Wasserstein space $({\mathcal{P}}_2(\mathbb{M}),W_2)$ of probability measures in a finite dimensional Euclidean space, a complete Riemannian manifold, or a separable Hilbert space $\mathbb{M}$. We will show that such a Sobolev space is always Hilbertian, independently of the choice of the reference measure $\mathfrak{m}$ so that the resulting Cheeger energy is a Dirichlet form. We will eventually provide an explicit characterization for the corresponding notion of $\mathfrak{m}$-Wasserstein gradient, showing useful calculus rules and its consistency with the tangent bundle and the $\mathrm{\Gamma }$-calculus inherited from the Dirichlet form.