Revista Matemática Iberoamericana

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Volume 35, Issue 7, 2019, pp. 2119–2150
DOI: 10.4171/rmi/1114

Published online: 2019-07-22

On Cheeger and Sobolev differentials in metric measure spaces

Martin Kell[1]

(1) Universität Tübingen, Germany

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 differentiability spaces are reflexive.

Keywords: Lipschitz functions, Sobolev functions, metric measure spaces

Kell Martin: On Cheeger and Sobolev differentials in metric measure spaces. Rev. Mat. Iberoam. 35 (2019), 2119-2150. doi: 10.4171/rmi/1114