A short proof of the infinitesimal Hilbertianity of the weighted Euclidean space
Di Marino, S., Lučić, D., & Pasqualetto, E. (2020). A short proof of the infinitesimal Hilbertianity of the weighted Euclidean space. Comptes Rendus Mathematique, 358(7), 817-825. https://doi.org/10.5802/crmath.88
Published inComptes Rendus Mathematique
© Académie des sciences, Paris and the authors, 2020
We provide a quick proof of the following known result: the Sobolev space associated with the Euclidean space, endowed with the Euclidean distance and an arbitrary Radon measure, is Hilbert. Our new approach relies upon the properties of the Alberti–Marchese decomposability bundle. As a consequence of our arguments, we also prove that if the Sobolev norm is closable on compactly-supported smooth functions, then the reference measure is absolutely continuous with respect to the Lebesgue measure.
PublisherInstitut de France
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Related funder(s)Academy of Finland
Funding program(s)Centre of Excellence, AoF; Research costs of Academy Research Fellow, AoF; Academy Project, AoF; Research post as Academy Research Fellow, AoF
Additional information about fundingThe second and third named authors acknowledge the support by the Academy of Finland, projects 274372, 307333, 312488, and 314789. The first named author is a member of GNAMPA, INdAM.
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