Infinitesimal Hilbertianity of weighted Riemannian manifolds

Abstract

The main result of this paper is the following: any `weighted' Riemannian manifold $(M,g,\mu )$ - i.e. endowed with a generic non-negative Radon measure $\mu$ - is `infinitesimally Hilbertian', which means that its associated Sobolev space $W^{1,2}(M,g,\mu )$ is a Hilbert space. We actually prove a stronger result: the abstract tangent module (à la Gigli) associated to any weighted reversible Finsler manifold $(M,F,\mu )$ can be isometrically embedded into the space of all measurable sections of the tangent bundle of $M$ that are $2$-integrable with respect to $\mu$.

References in corpus (4)

• Metric measure spaces with Riemannian Ricci curvature bounded from below
• Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below
• Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces
• Riemannian Ricci curvature lower bounds in metric measure spaces with $\sigma$-finite measure

Cited by in corpus (2)

• Generalized Bakry-Émery curvature condition and equivalent entropic inequalities in groups
• Metric Sobolev spaces I: equivalence of definitions