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Nirenberg's famous complex Frobenius theorem gives necessary and sufficientconditions on a locally integrable structure for when the manifold is locallydiffeomorphic 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 locallyspanned by $F^*\frac\partial{\partial t^1},\dots,F^*\frac\partial{\partialt^r},F^*\frac\partial{\partial z^1},\dots,F^*\frac\partial{\partial z^m}$,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-Zygmundregularity for the coordinate charts which achieve this realization. Namely, ifthe structure has H\"older-Zygmund regularity of order $\alpha>1$, then thecoordinate chart $F$ that maps to $\mathbb R^r\times\mathbb C^m \times\mathbbR^{N-r-2m}$ 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 thevector fields $F^*\frac\partial{\partial t^1},\dots,F^*\frac\partial{\partialt^r},F^*\frac\partial{\partial z^1},\dots,F^*\frac\partial{\partial z^m}$ onthe original manifold have H\"older-Zygmund regularity of order$\alpha-\varepsilon$ for every $\varepsilon>0$, and we give an example to showthat the regularity for $F^*\frac\partial{\partial z}$ is optimal.

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