Integral representations for bracket-generating multi-flows

If f1, f2 are smooth vector fields on an open subset of an Euclidean space and [f1, f2] is their Lie bracket, the asymptotic formula (Equation presented), (1) where we have set Φ[f1 f2](t1,t2)(x)def= exp(-t2f2) oexp(-t1f1) o exp(t2f2)° exp(t1f1)(x), is valid for all t1,t2 small enough. In fact, the integral, exact formula (Equation presented), (2) where [f1, f2](s2,s1)(y)def= D exp(s1f1) oexp(s2f2))-1(y) · [f1, f2](exp(s1f1) o exp(s2f2)(y)), has also been proven. Of course (2) can be regarded as an improvement of (1). In this paper we show that an integral representation like (2) holds true for any iterated Lie bracket made of elements of a family f1,..., fm of vector fields. In perspective, these integral representations might lie at the basis for extensions of asymptotic formulas involving non-smooth vector fields.