Calculus of Variations and Geometric Measure Theory

E. Le Donne - D. Lučić - E. Pasqualetto

Universal infinitesimal Hilbertianity of sub-Riemannian manifolds

created by pasqualetto on 15 Oct 2019
modified on 07 Nov 2021


Accepted Paper

Inserted: 15 oct 2019
Last Updated: 7 nov 2021

Journal: Potential Analysis
Year: 2019

ArXiv: 1910.05962 PDF


We prove that sub-Riemannian manifolds are infinitesimally Hilbertian (i.e., the associated Sobolev space is Hilbert) when equipped with an arbitrary Radon measure. The result follows from an embedding of metric derivations into the space of square-integrable sections of the horizontal bundle, which we obtain on all weighted sub-Finsler manifolds. As an intermediate tool, of independent interest, we show that any sub-Finsler distance can be monotonically approximated from below by Finsler ones. All the results are obtained in the general setting of possibly rank-varying structures.

Tags: GeoMeG