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

Di Marino, Simone; Lučić, Danka; Pasqualetto, Enrico

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

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.

Classification : 53C23, 46E35, 26B05

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