Abstract: In this talk, we present a general criterium for the density in energy of suitable subalgebras of Lipschitz functions in the q-metric-Sobolev space associated with a Polish metric-measure space.

Motivated by the growing interest in partial differential equations on spaces of probability measures and by applications to Data Science, we apply our result to the case of the algebra of cylindrical functions in the q-Wasserstein-Sobolev space arising from a positive Borel measure on the p-Kantorivich-Rubinstein-Wasserstein space of probability measures on a complete and separable metric space.

As an application, we prove that, in case the underlying metric space is a separable Banach space, many of its properties pass to the Wasserstein-Sobolev space: this is the case of Hilbertianity, reflexivity, uniform convexity and Clarkson’s type inequalities.

This talk is partly based on a joint work with Massimo Fornasier (TU München, Germany) and Giuseppe Savaré (Bocconi University, Milano, Italy).