Improving the regularity of vector fields

Let α>0, β>α, and let X1,…,Xq be Clocα vector fields on a Cα+1 manifold which span the tangent space at every point, where Cs denotes the Zygmund-Hölder space of order s. We give necessary and sufficient conditions for when there is a Cβ+1 structure on the manifold, compatible with its Cα+1 structure, with respect to which X1,…,Xq are Clocβ. This strengthens previous results of the first author which dealt with the setting α>1, β>max⁡{α,2}.