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arXiv · 1812.02086

Infinitesimal Hilbertianity of locally CAT($\kappa$)-spaces

Simone Di Marino·Nicola Gigli·Enrico Pasqualetto·Elefterios Soultanis
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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 $\mu$ on $Y$ giving finite mass to bounded sets, the resulting metric measure space $(Y,d,\mu)$ is infinitesimally Hilbertian, i.e. the Sobolev space $W^{1,2}(Y,d,\mu)$ 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 $x\in Y$ is the tangent cone at $x$ of $Y$. The conclusion then follows from the fact that for every $x\in Y$ such a cone is a CAT(0)-space and, as such, has a Hilbert-like structure.

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Simone Di Marino, Nicola Gigli, Enrico Pasqualetto, Elefterios Soultanis. 2018-12-05. Infinitesimal Hilbertianity of locally CAT($\kappa$)-spaces. https://arxiv.org/abs/1812.02086

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