A1 Journal article (refereed)
A short proof of the infinitesimal Hilbertianity of the weighted Euclidean space (2020)
Une preuve courte de la Hilbertianité infinitésimale de l’espace euclidien à poids

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

JYU authors or editors

Lucic, Danka
Pasqualetto, Enrico

Publication details

All authors or editors: Di Marino, Simone; Lučić, Danka; Pasqualetto, Enrico

Journal or series: Comptes Rendus Mathematique

ISSN: 1631-073X

eISSN: 1778-3569

Publication year: 2020

Volume: 358

Issue number: 7

Pages range: 817-825

Publisher: Institut de France

Publication country: France

Publication language: English

DOI: https://doi.org/10.5802/crmath.88

Publication open access: Openly available

Publication channel open access: Open Access channel

Publication is parallel published (JYX): https://jyx.jyu.fi/handle/123456789/73342

Web address of parallel published publication (pre-print): https://arxiv.org/abs/2005.02924

Abstract

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.

Keywords: differential geometry; functional analysis

Fields of science: