Frobenius theorem and fine structure of tangency sets to non-involutive distributions

Abstract

In this paper we provide a complete answer to the question whether Frobenius' Theorem can be generalized to surfaces below the $C^{1,1}$ threshold. We study the fine structure of the tangency set in terms of involutivity of a given distribution and we highlight a tradeoff behavior between the regularity of a tangent surface and that of the tangency set. First of all, we prove a Frobenius-type result, that is, given a $k$-dimensional surface $S$ of class $C^1$ and a non-involutive $k$-distribution $V$, if $E$ is a Borel set contained in the tangency set $\ddot{I}„(S,V)$ of $S$ to $V$ and $1_E\in W^{s,1}(S)$ with $s>1/2$ then $E$ must be ${\mathcal{H}}^k$-null in $S$. In addition, if $S$ is locally a graph of a $C^1$ function with gradient in $W^{\hat{I}\pm ,q}$ and if a Borel set $E\subset \ddot{I}„(S,V)$ satisfies $1_E\in W^{s,1}(S)$ with \[ s \in \bigl(0,\tfrac{1}{2}\bigr]\qquad\text{and}\qquadα\;>\; 1 - \Bigl(2 - \tfrac{1}{q}\Bigr) \, s, \] then ${\mathcal{H}}^k(E)=0$. We show this exponents' condition to be sharp by constructing, for any $\hat{I}\pm <1-(2-\frac{1}{q})s$, a surface $S$ in the same class as above and a set $E\subset \ddot{I}„(S,V)$ with $1_E\in W^{s,1}(S)$ and ${\mathcal{H}}^k(E)>0$. Our methods combine refined fractional Sobolev estimates on rectifiable sets, a Stokes-type theorem for rough forms on finite-perimeter sets, and a generalization of the Lusin's Theorem for gradients.