Improving the Regularity of Vector Fields

• Main Authors: Street, Brian, Yao, Liding
• Format: Journal Article
• Language: English
• Published: 22-05-2022
• Subjects: Mathematics - Classical Analysis and ODEs; Mathematics - Differential Geometry
• DOI: 10.48550/arxiv.2105.10120

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