Improving the Regularity of Vector Fields

Let $α>0$, $β>α$, and let $X_1,\ldots, X_q$ be $\mathscr{C}^α_{\mathrm{loc}}$ vector fields on a $\mathscr{C}^{α+1}$ manifold which span the tangent space at every point, where $\mathscr{C}^{s}$ denotes the Zygmund-Hölder space of order $s$. We give necessary and sufficient conditions for when there is a $\mathscr{C}^{β+1}$ structure on the manifold, compatible with its $\mathscr{C}^{α+1}$ structure, with respect to which $X_1,\ldots, X_q$ are $\mathscr{C}^β_{\mathrm{loc}}$. This strengthens previous results of the first author which dealt with the setting $α>1$, $β>\max\{ α, 2\}$.

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