A short proof of the infinitesimal Hilbertianity of the weighted Euclidean space

Simone Di Marino; Danka Lučić; Enrico Pasqualetto

doi:10.5802/crmath.88

Théorie des fonctions et espaces des fonctions

A short proof of the infinitesimal Hilbertianity of the weighted Euclidean space
[Une preuve courte de la Hilbertianité infinitésimale de l’espace euclidien à poids]

Simone Di Marino ¹ ; Danka Lučić ² ; Enrico Pasqualetto ²

¹ Dipartimento di Matematica (DIMA), Via Dodecaneso 35, 16146 Genova, Università di Genova, Italy
² Department of Mathematics and Statistics, P.O. Box 35 (MaD), 40014 University of Jyvaskyla, Finland

Comptes Rendus. Mathématique, Volume 358 (2020) no. 7, pp. 817-825.

Résumé
Abstract

Nous fournissons une preuve courte du résultat connu suivant : l’espace de Sobolev associé à l’espace euclidien muni de sa distance euclidienne et d’une mesure arbitraire de Radon, est un espace d’Hilbert. Notre nouvelle approche repose sur des propriétés du fibré de décomposabilité introduit par Alberti et Marchese. En conséquence de nos arguments, nous prouvons aussi que si la norme de Sobolev est fermable dans les fonctions lisses à support compact, la mesure de référence est absolument continue par rapport à la mesure de Lebesgue.

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.

Reçu le : 2020-05-06
Révisé le : 2020-06-18
Accepté le : 2020-06-19
Publié le : 2020-11-16

DOI : 10.5802/crmath.88

Classification : 53C23, 46E35, 26B05

Affiliations des auteurs :

Simone Di Marino ¹ ; Danka Lučić ² ; Enrico Pasqualetto ²

¹ Dipartimento di Matematica (DIMA), Via Dodecaneso 35, 16146 Genova, Università di Genova, Italy
² Department of Mathematics and Statistics, P.O. Box 35 (MaD), 40014 University of Jyvaskyla, Finland

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Simone Di Marino and Danka Lu\v{c}i\'c and Enrico Pasqualetto},
     title = {A short proof of the infinitesimal {Hilbertianity} of the weighted {Euclidean} space},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {817--825},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {7},
     year = {2020},
     doi = {10.5802/crmath.88},
     language = {en},
}

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Simone Di Marino; Danka Lučić; Enrico Pasqualetto. A short proof of the infinitesimal Hilbertianity of the weighted Euclidean space. Comptes Rendus. Mathématique, Volume 358 (2020) no. 7, pp. 817-825. doi : 10.5802/crmath.88. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.88/

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