TITLE: Frobenius Theorem for subbundle of low regularity?
QUESTION [13 upvotes]: Frobenius Theorem says that a subbundle $E$ of the tangent bundle $TM$ of a manifold $M$ is tangent to a foliation if and only if for any two vector fields $X, Y \subset E$ the bracket $[X,Y]\subset E$. Bracket is a second order operator, hence subbundle $E$ needs to be $C^2$.
Are there any generalizations for subbundle which is $C^1$, $C^{1+smth}$?
Thank you,
Z.

REPLY [16 votes]: You can even get a Frobenius theorem for Lipshitz vector fields,
which need not  span everywhere (the rank can drop along closed subsets).
The state of the art in this domain is done by the control theorists, particularly
Sussmann and Agrachev.  Here is an abstract for a recent article which will have
other earlier references:
Journal of Differential Equations
Volume 243, Issue 2, 15 December 2007, Pages 270-300. F. Rampazzo and H.J. Sussmann
Abstract: We generalize the classical Frobenius Theorem to distributions that are spanned by locally Lipschitz vector fields. The various versions of the involutivity conditions are extended by means of set-valued Lie derivatives—in particular, set-valued Lie brackets—and set-valued exterior derivatives. A PDEs counterpart of these Frobenius-type results is investigated as well.