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 $\varepsilon>0$, locally there is a homeomorphism $Φ(u,v)$ such that $Φ,\frac{\partialΦ}{\partial u^1},\dots,\frac{\partialΦ}{\partial u^r}\in C^{0,1-\varepsilon}$, and $\mathcal V$ is spanned by the continuous vector fields $Φ_*\frac\partial{\partial u^1},\dots,Φ_*\frac\partial{\partial u^r}$.