Improving the Regularity of Vector Fields

TL;DR

This paper establishes necessary and sufficient conditions for enhancing the regularity of vector fields on manifolds, allowing for a compatible higher-regularity structure that makes the vector fields smoother.

Contribution

It extends previous regularity results to cases where the initial vector fields have lower regularity, specifically in Zygmund-H"older spaces, providing a broader understanding of manifold structures.

Findings

Characterization of when a higher-regularity structure exists

Extension of regularity results to lower initial regularity

Conditions for compatibility of manifold structures with 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 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 $\alpha>1$, $\beta>\max\{ \alpha, 2\}$.