Step-s involutive families of vector fields, their orbits and the Poincar\'e inequality

TL;DR

This paper investigates families of smooth vector fields satisfying higher order involutivity, exploring their commutators, the structure of their orbits, and establishing a Poincaré inequality relevant to their geometric properties.

Contribution

It introduces a framework for analyzing involutive vector fields of higher order, clarifies the regularity of their orbits, and proves a Poincaré inequality in this context.

Findings

Definition of higher order involutivity for vector fields

Regularity results for Sussmann's orbits

Establishment of a Poincaré inequality for these vector fields

Abstract

We consider a family of $C^1$ vector fields satisfying a suitable higher order involutivity condition. We discuss the definition of commutators, the regularity of Sussmann's orbits and the Poincar\'e inequality.