The Frobenius Theorem for Log-Lipschitz Subbundles

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 $\mathrm{\Phi }(u,v)$ such that $\Phi,\frac{\partial\Phi}{\partial u^1},\dots,\frac{\partial\Phi}{\partial u^r}\in C^{0,1-\varepsilon}$,and$\mathcal V$ is spanned by the continuous vector fields $\Phi_\frac\partial{\partial u^1},\dots,\Phi_\frac\partial{\partial u^r}$.