On Cheeger and Sobolev differentials in metric measure spaces

2015/12/02 by Martin Kell, Kell, Martin
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG) #math.FA #math.MG

paper · pdf · doi:10.48550/arxiv.1512.00828

Comments welcome!

arxiv created 2015/12/02 · arxiv updated 2015/12/03

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.

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