Comptes Rendus
no. 7
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 Marino1  ; Danka Lučić2 ; Enrico Pasqualetto2
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.

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.

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DOI : 10.5802/crmath.88
Classification : 53C23, 46E35, 26B05
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},
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     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/

[1] Giovanni Alberti; Andrea Marchese On the differentiability of Lipschitz functions with respect to measures in the Euclidean space, Geom. Funct. Anal., Volume 26 (2016) no. 1, pp. 1-66 | DOI | MR | Zbl

[2] Luigi Ambrosio; Nicola Gigli; Giuseppe Savaré Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics, ETH Zürich, Birkhäuser, 2008 | Zbl

[3] Luigi Ambrosio; Nicola Gigli; Giuseppe Savaré Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces, Rev. Mat. Iberoam., Volume 29 (2013) no. 3, pp. 969-996 | DOI | MR | Zbl

[4] Luigi Ambrosio; Nicola Gigli; Giuseppe Savaré Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below, Invent. Math., Volume 195 (2014) no. 2, pp. 289-391 | DOI | MR | Zbl

[5] Vladimir I. Bogachev Differentiable Measures and the Malliavin Calculus, Mathematical Surveys and Monographs, 164, American Mathematical Society, 2010 | MR | Zbl

[6] Jeff Cheeger Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal., Volume 9 (1999) no. 3, pp. 428-517 | DOI | MR | Zbl

[7] Simone Di Marino Recent advances on BV and Sobolev Spaces in metric measure spaces, Ph. D. Thesis, Scuola Normale Superiore di Pisa (Italy) (2014)

[8] Simone Di Marino; Nicola Gigli; Enrico Pasqualetto; Elefterios Soultanis Infinitesimal Hilbertianity of locally CAT (κ)-spaces (2018) (https://arxiv.org/abs/1812.02086, submitted)

[9] Nicola Gigli On the differential structure of metric measure spaces and applications, Memoirs of the American Mathematical Society, 236, American Mathematical Society, 2015, vi+91 pages | MR | Zbl

[10] Nicola Gigli; Enrico Pasqualetto Behaviour of the reference measure on RCD spaces under charts (2016) (https://arxiv.org/abs/1607.05188, to appear in Commun. Anal. Geom.)

[11] Danka Lučić; Enrico Pasqualetto Infinitesimal Hilbertianity of weighted Riemannian manifolds, Can. Math. Bull., Volume 63 (2020) no. 1, pp. 118-140 | DOI | MR | Zbl

[12] Guido De Philippis; Filip Rindler On the structure of 𝒜-free measures and applications, Ann. Math., Volume 184 (2016) no. 3, pp. 1017-1039 | DOI | MR | Zbl

[13] Nageswari Shanmugalingam Newtonian spaces: an extension of Sobolev spaces to metric measure spaces, Rev. Mat. Iberoam., Volume 16 (2000) no. 2, pp. 243-279 | DOI | MR | Zbl

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