The Frobenius Theorem for Log-Lipschitz Subbundles

Abstract

We extend the definition of involutivity to non-Lipschitz tangent subbundles using generalized functions. We prove the Frobenius Theorem with sharp regularity estimate when the subbundle is log-Lipschitz: if $\mathcal{V}$ is a log-Lipschitz involutive subbundle of rank $r$, then for any $\epsilon >0$, locally there is a homeomorphism $\Phi (u,v)$ such that $\Phi ,\frac{\partial \Phi }{\partial u^1},\dots ,\frac{\partial \Phi }{\partial u^r}\in C^{0,1-\epsilon }$, and $\mathcal{V}$ is spanned by the continuous vector fields ${\Phi }_{\ast }\frac{\partial }{\partial u^1},\dots ,{\Phi }_{\ast }\frac{\partial }{\partial u^r}$.

arXiv commentFinal version, 33 pages. Appeared in The Journal of Geometric Analysis