Nirenberg’s famous complex Frobenius theorem gives necessary and sufficient conditions on a locally integrable structure for when the manifold is locally diffeomorphic to ℝr × ℂm × ℝN−r−2m through a coordinate chart F in such a way that the structure is locally spanned by , where we have given ℝr × ℂm × ℝN−r−2m coordinates (t, z, s). In this paper, we give the optimal Hölder–Zygmund regularity for the coordinate charts which achieve this realization. Namely, if the structure has Hölder–Zygmund regularity of order α > 1, then the coordinate chart F that maps to ℝr × ℂm × ℝN−r−2m may be taken to have Hölder–Zygmund regularity of order α, and this is sharp. Furthermore, we can choose this F in such a way that the vector fields on the original manifold have Hölder–Zygmund regularity of order α − ε for every ε > 0, and we give an example to show that the regularity for \({F^\ast}{\partial \over {\partial z}}\) is optimal.
