Sharp Hölder Regularity for Nirenberg's Complex Frobenius Theorem

Abstract

Nirenberg's famous complex Frobenius theorem gives necessary and sufficient conditions on a locally integrable structure for when the manifold is locally diffeomorphic to ${\mathbb{R}}^r\times {\mathbb{C}}^m\times {\mathbb{R}}^{N-r-2m}$ through a coordinate chart $F$ in such a way that the structure is locally spanned by $F^{\ast }\frac{\partial }{\partial t^1},\dots ,F^{\ast }\frac{\partial }{\partial t^r},F^{\ast }\frac{\partial }{\partial z^1},\dots ,F^{\ast }\frac{\partial }{\partial z^m}$, where we have given ${\mathbb{R}}^r\times {\mathbb{C}}^m\times {\mathbb{R}}^{N-r-2m}$ 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 $\alpha >1$, then the coordinate chart $F$ that maps to ${\mathbb{R}}^r\times {\mathbb{C}}^m\times {\mathbb{R}}^{N-r-2m}$ may be taken to have Hölder-Zygmund regularity of order $\alpha$, and this is sharp. Furthermore, we can choose this $F$ in such a way that the vector fields $F^{\ast }\frac{\partial }{\partial t^1},\dots ,F^{\ast }\frac{\partial }{\partial t^r},F^{\ast }\frac{\partial }{\partial z^1},\dots ,F^{\ast }\frac{\partial }{\partial z^m}$ on the original manifold have Hölder-Zygmund regularity of order $\alpha -\epsilon$ for every $\epsilon >0$, and we give an example to show that the regularity for $F^{\ast }\frac{\partial }{\partial z}$ is optimal.

68 pages, including 10 pages of appendix