We show that, given a metric space (Y , d) of curvature bounded from above in the sense of Alexandrov, and a positive Radon measure μ on Y giving finite mass to bounded sets, the resulting metric measure space (Y , d, μ) is infinitesimally Hilbertian, i.e. the Sobolev space W1 , 2(Y , d, μ) is a Hilbert space. The result is obtained by constructing an isometric embedding of the ‘abstract and analytical’ space of derivations into the ‘concrete and geometrical’ bundle whose fibre at x∈ Y is the tangent cone at x of Y. The conclusion then follows from the fact that for every x∈ Y such a cone is a CAT (0) space and, as such, has a Hilbert-like structure.

Infinitesimal Hilbertianity of Locally CAT (κ) -Spaces / Di Marino, S.; Gigli, N.; Pasqualetto, E.; Soultanis, E.. - In: THE JOURNAL OF GEOMETRIC ANALYSIS. - ISSN 1050-6926. - (In corso di stampa), pp. 1-45. [10.1007/s12220-020-00543-7]

Infinitesimal Hilbertianity of Locally CAT (κ) -Spaces

Gigli N.
;
Pasqualetto E.;Soultanis E.
In corso di stampa

Abstract

We show that, given a metric space (Y , d) of curvature bounded from above in the sense of Alexandrov, and a positive Radon measure μ on Y giving finite mass to bounded sets, the resulting metric measure space (Y , d, μ) is infinitesimally Hilbertian, i.e. the Sobolev space W1 , 2(Y , d, μ) is a Hilbert space. The result is obtained by constructing an isometric embedding of the ‘abstract and analytical’ space of derivations into the ‘concrete and geometrical’ bundle whose fibre at x∈ Y is the tangent cone at x of Y. The conclusion then follows from the fact that for every x∈ Y such a cone is a CAT (0) space and, as such, has a Hilbert-like structure.
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Di Marino, S.; Gigli, N.; Pasqualetto, E.; Soultanis, E.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/118311
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