Revista Matemática Iberoamericana
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Published online: 2019-07-22
On Cheeger and Sobolev differentials in metric measure spacesMartin Kell (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