On Cheeger and Sobolev differentials in metric measure spaces

TL;DR

This paper integrates Gigli's Sobolev calculus with Cheeger's differentiability spaces, demonstrating the finite dimensionality of Lipschitz modules and reflexivity of Sobolev spaces in these non-smooth metric measure spaces.

Contribution

It shows the compatibility of Gigli's Sobolev calculus with differentiability space theory and establishes finite dimensionality and reflexivity results for Sobolev spaces in these settings.

Findings

Lipschitz modules are pointwise finite dimensional in differentiability spaces.

The Sobolev spaces in these spaces are reflexive.

A relaxation procedure for $L^p$-valued subadditive functionals is introduced.

Abstract

Recently Gigli developed a Sobolev calculus on non-smooth spaces using module theory. In this paper it is shown that his theory fits nicely into the theory of differentiability spaces initiated by Cheeger, Keith and others. A relaxation procedure for $L^p$-valued subadditive functionals is presented and a relationship between the module generated by a functional and the one generated by its relaxation is given. In the framework of differentiability spaces, which includes 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.