6533b7cefe1ef96bd1257a77

RESEARCH PRODUCT

Universal infinitesimal Hilbertianity of sub-Riemannian manifolds

Enrico Le DonneDanka LučićEnrico Pasqualetto

subject

Mathematics - Differential GeometryMetric Geometry (math.MG)Sobolev spaceFunctional Analysis (math.FA)Mathematics - Functional AnalysisRiemannin monistotdifferentiaaligeometriasub-Finsler manifoldMathematics - Metric GeometryDifferential Geometry (math.DG)infinitesimal hilbertianityFOS: MathematicsMathematics::Metric Geometrysub-Riemannian manifoldMathematics::Differential GeometrymonistotfunktionaalianalyysiMathematics::Symplectic Geometry53C23 46E35 53C17 55R25Analysis

description

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.

yearjournalcountryeditionlanguage
2019-10-14
https://dx.doi.org/10.48550/arxiv.1910.05962