  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 $\Phi(u,v)$ 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 $\Phi_*\frac\partial{\partial
u^1},\dots,\Phi_*\frac\partial{\partial u^r}$.
