Improving the Regularity of Vector Fields

Abstract

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

53 pages. Final version, to be appeared in Journal of Functional Analysis

References in corpus (1)

• Regularity Theory for Elliptic PDE