Infinitesimal Hilbertianity of Weighted Riemannian Manifolds

Abstract

The main result of this paper is the following: any weighted Riemannian manifold (M,g,𝜇), i.e., a Riemannian manifold (M,g) endowed with a generic non-negative Radon measure 𝜇, is infinitesimally Hilbertian, which means that its associated Sobolev space W1,2(M,g,𝜇) 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,𝜇) can be isometrically embedded into the space of all measurable sections of the tangent bundle of M that are 2-integrable with respect to 𝜇.

By following the same approach, we also prove that all weighted (sub-Riemannian) Carnot groups are infinitesimally Hilbertian.