On Cheeger and Sobolev differentials in metric measure spaces

• Martin Kell ^[1]
  1. [1] Universität Tübingen
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 35, Nº 7, 2019, págs. 2119-2150
• Idioma: inglés
• DOI: 10.4171/rmi/1114
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• Resumen

  • 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 Lp-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.