Step-s involutive families of vector fields, their orbits and the Poincaré inequality

We consider a family H:={X1,…,Xm} of vector fields in Rn. Under a suitable s-involutivity assumption on commutators of order at most s, we show a ball-box theorem for Carnot–Carathéodory balls of the family H and we prove the related Poincaré inequality. Each control ball is contained in a suitable Sussmannʼs orbit of which we discuss some regularity properties. Our main tool is a class of almost exponential maps which we discuss carefully under low regularity assumptions on the coefficients of the vector fields in H.