Geodesic
Infinitesimal Hilbertianity of Weighted Riemannian Manifolds
Canadian mathematical bulletin, Tome 63 (2020) no. 1, pp. 118-140

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DOI

The main result of this paper is the following: any weighted Riemannian manifold $(M,g,\unicode[STIX]{x1D707})$, i.e., a Riemannian manifold $(M,g)$ endowed with a generic non-negative Radon measure $\unicode[STIX]{x1D707}$, is infinitesimally Hilbertian, which means that its associated Sobolev space $W^{1,2}(M,g,\unicode[STIX]{x1D707})$ is a Hilbert space.We actually prove a stronger result: the abstract tangent module (à la Gigli) associated with any weighted reversible Finsler manifold $(M,F,\unicode[STIX]{x1D707})$ can be isometrically embedded into the space of all measurable sections of the tangent bundle of $M$ that are $2$-integrable with respect to $\unicode[STIX]{x1D707}$.By following the same approach, we also prove that all weighted (sub-Riemannian) Carnot groups are infinitesimally Hilbertian.
DOI : 10.4153/S0008439519000328
Mots-clés : infinitesimal Hilbertianity, Sobolev space, Finsler manifold, smooth approximation of Lipschitz functions 
Lučić, Danka; Pasqualetto, Enrico. Infinitesimal Hilbertianity of Weighted Riemannian Manifolds. Canadian mathematical bulletin, Tome 63 (2020) no. 1, pp. 118-140. doi: 10.4153/S0008439519000328
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     author = {Lu\v{c}i\'c, Danka and Pasqualetto, Enrico},
     title = {Infinitesimal {Hilbertianity} of {Weighted} {Riemannian} {Manifolds}},
     journal = {Canadian mathematical bulletin},
     pages = {118--140},
     year = {2020},
     volume = {63},
     number = {1},
     doi = {10.4153/S0008439519000328},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008439519000328/}
}
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