JournalsrmiVol. 35, No. 7pp. 2119–2150

On Cheeger and Sobolev differentials in metric measure spaces

  • Martin Kell

    Universität Tübingen, Germany
On Cheeger and Sobolev differentials in metric measure spaces cover
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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 LpL^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,NK,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.

Cite this article

Martin Kell, On Cheeger and Sobolev differentials in metric measure spaces. Rev. Mat. Iberoam. 35 (2019), no. 7, pp. 2119–2150

DOI 10.4171/RMI/1114