Simple proof of the theorem on the lifting of vector fields

Abstract

LetX1, . . . ,Xn be an n-tuple of vector fields of class Cr−1, r ≥ 1, in a neighborhood U ⊂ RN of the origin. For each multi-index I = (i1, . . . , ik), where 1 ≤ il ≤ n, l = 1, . . . , k, we define the order |I| = k. To each multi-index I, |I| = k ≤ r, we assign the kth-order differential operator XI = Xi1Xi2 · · ·Xik and the iterated commutator X[I] = [Xi1 , [Xi2 , . . . [Xik−1 ,Xik ] . . . ]], where [ · , · ] is the Lie bracket. The operators XI and X[I] belong to the smoothness class Cr−|I|. Note that each commutatorX[I] can be represented as a linear combination of operators of order |I|, X[I] = ∑