Step-\(s\) involutive families of vector fields, their orbits and the Poincaré inequality (Q387843)

The authors extend basic machinery of sub-Riemannian geometric analysis to families of vector fields lacking the requisite smoothness necessary for application of Hörmander's classical theorem. They consider families of Euclidean vector fields of class \(C^s\) whose commutators satisfy (somewhat technical) regularity hypotheses. For such families, they establish (i) a ball-box theorem for a suitable almost exponential map, (ii) a doubling property for the Lebesgue measure of control balls, and (iii) an appropriate Poincaré inequality. A key technical complication in their arguments involves injectivity of the almost exponential map, which cannot be obtained by standard methods.