Infinitesimal Hilbertianity of Weighted Riemannian Manifolds

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...

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Bibliographic Details
Published in:Canadian mathematical bulletin 2020-03, Vol.63 (1), p.118-140
Main Authors: Lučić, Danka, Pasqualetto, Enrico
Format: Article
Language:English
Subjects:
Algebra
Geometry
Hilbert space
Mathematics
Riemann manifold
Sobolev space
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Summary: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.
ISSN:0008-4395
1496-4287
DOI:10.4153/S0008439519000328