On Cheeger and Sobolev differentials in metric measure spaces

Recently Gigli developed a Sobolev calculus on non-smooth spaces using module theory. He shows that the relaxed notion of gradient is sufficient to obtain “1-forms” and make it possible to define Sobolev differentials which resemble the ones in the smooth setting.

In this talk I will show that his theory fits nicely into the theory of Lipschitz differentiable spaces initiated by Cheeger, Keith and others. For this I present a new relaxation procedure for $L^p$-valued subadditive functionals and give a relationship between the module generated by a functional and the module generated by its relaxation.

In the framework of Lipschitz differentiable spaces, which include so called PI- and RCD(K,N)-spaces, the Lipschitz module is “pointwise” finite dimensional. A general renorming theorem together with the characterization above shows that the Sobolev spaces of such spaces are reflexive.