Generalized Jacobi identities and ball-box theorem for horizontally regular vector fields

TL;DR

This paper establishes generalized Jacobi identities and a ball-box theorem for vector fields with horizontal regularity, extending classical results to nonsmooth settings relevant for sub-Riemannian geometry.

Contribution

It introduces a unified notion of commutator for nonsmooth vector fields and proves key geometric inequalities under horizontal regularity assumptions.

Findings

Generalized Jacobi identities for nonsmooth vector fields

Ball-box theorem for nonsmooth H"ormander vector fields

Poincaré inequality in the nonsmooth setting

Abstract

We consider a family of vector fields and we assume a horizontal regularity on their derivatives. We discuss the notion of commutator showing that different definitions agree. We apply our results to the proof of a ball-box theorem and Poincar\'e inequality for nonsmooth H\"ormander vector fields.