Sharp H\"older Regularity for Nirenberg's Complex Frobenius Theorem

TL;DR

This paper establishes the optimal H"older-Zygmund regularity for coordinate charts in Nirenberg's complex Frobenius theorem, showing that the regularity of the structure and charts are closely matched and sharp.

Contribution

It provides the sharp H"older-Zygmund regularity results for coordinate charts realizing Nirenberg's complex Frobenius structures, extending the regularity theory.

Findings

Coordinate charts can be taken with H"older-Zygmund regularity of order lpha when the structure has regularity lpha.

The regularity of the vector fields on the original manifold can be made arbitrarily close to lpha.

The regularity for the vector fields associated with the complex structure is shown to be optimal.

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 $F^*\frac\partial{\partial t^1},\dots,F^*\frac\partial{\partial t^r},F^*\frac\partial{\partial z^1},\dots,F^*\frac\partial{\partial z^m}$, 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 H\"older-Zygmund regularity for the coordinate charts which achieve this realization. Namely, if the structure has H\"older-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 H\"older-Zygmund regularity of order $\alpha$, and this is sharp. Furthermore, we can choose this $F$ in such a way that the vector fields $F^*\frac\partial{\partial t^1},\dots,F^*\frac\partial{\partial t^r},F^*\frac\partial{\partial z^1},\dots,F^*\frac\partial{\partial z^m}$ on the original manifold have H\"older-Zygmund regularity of order $\alpha-\varepsilon$ for every $\varepsilon>0$, and we give an example to show that the regularity for $F^*\frac\partial{\partial z}$ is optimal.