# Infinitesimal Hilbertianity of Locally $$\mathrm{CAT}(\kappa )$$-Spaces

• Di Marino, Simone Istituto Nazionale di Alta Matematica, Unità INdAM SNS Pisa, Piazza dei Cavalieri 7, 56126 Pisa, Italy
• Gigli, Nicola SISSA, Via Bonomea 265, 34136 Trieste, Italy
• Pasqualetto, Enrico SISSA, Via Bonomea 265, 34136 Trieste, Italy - Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), 40014 Jyväskylä, Finland
06.11.2020
Published in:
• The Journal of Geometric Analysis. - 2020
English We show that, given a metric space $$(\mathrm{Y},\textsf {d} )$$of curvature bounded from above in the sense of Alexandrov, and a positive Radon measure $$\mu$$on $$\mathrm{Y}$$giving finite mass to bounded sets, the resulting metric measure space $$(\mathrm{Y},\textsf {d} ,\mu )$$is infinitesimally Hilbertian, i.e. the Sobolev space $$W^{1,2}(\mathrm{Y},\textsf {d} ,\mu )$$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 $$x\in \mathrm{Y}$$is the tangent cone at x of $$\mathrm{Y}$$. The conclusion then follows from the fact that for every $$x\in \mathrm{Y}$$such a cone is a $$\mathrm{CAT}(0)$$space and, as such, has a Hilbert-like structure.
Faculty
Faculté des sciences et de médecine
Department
Département de Mathématiques
Language
• English
Classification
Mathematics