Improving the Regularity of Vector Fields
Abstract
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 ZygmundHölder 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 $\alpha>1$, $\beta>\max\{ \alpha, 2\}$.
 Publication:

arXiv eprints
 Pub Date:
 May 2021
 DOI:
 10.48550/arXiv.2105.10120
 arXiv:
 arXiv:2105.10120
 Bibcode:
 2021arXiv210510120S
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 Mathematics  Differential Geometry;
 2020: 57R25 (Primary);
 57R25;
 35R05 and 53C17 (Secondary)
 EPrint:
 53 pages. Final version, to be appeared in Journal of Functional Analysis