Universal infinitesimal Hilbertianity of sub-Riemannian manifolds

TL;DR

This paper demonstrates that sub-Riemannian manifolds equipped with any Radon measure have a Hilbertian Sobolev space structure, extending the understanding of their geometric and analytic properties.

Contribution

It establishes the infinitesimal Hilbertianity of sub-Riemannian manifolds with arbitrary measures and introduces a novel embedding of metric derivations into square-integrable sections.

Findings

Sub-Riemannian manifolds are infinitesimally Hilbertian with any Radon measure.

Any sub-Finsler distance can be approximated from below by Finsler distances.

The results hold for structures with possibly varying rank.

Abstract

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.