Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Sharp H\"older Regularity for Nirenberg's Complex Frobenius Theorem   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.

16
The categories of this paper are: stat
20
The categories of this paper are: acc-phys
6
The categories of this paper are: astro-ph
4
The categories of this paper are: cond-mat
23
The categories of this paper are: solv-int
15
The categories of this paper are: nucl-ex
36
The categories of this paper are: cmp-lg
21
The categories of this paper are: adap-org
0
The categories of this paper are: hep-ph
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The categories of this paper are: hep-lat
27
The categories of this paper are: comp-gas
30
The categories of this paper are: q-alg
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The categories of this paper are: econ
3
The categories of this paper are: physics
37
The categories of this paper are: supr-con
22
The categories of this paper are: chao-dyn