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

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 $\tau (S,V)$ of $S$ to $V$ and ${\mathbb{1}}_E\in W^{s,1}(S)$ with $s>1\mathrm{/}2$ 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 $W^{\alpha ,q}$ and if a Borel set $E\subset \tau (S,V)$ satisfies ${\mathbb{1}}_E\in W^{s,1}(S)$ with $$s\in (0,\frac{1}{2}]\text{and}\alpha >1-(2-\frac{1}{q})s,$$ then ${\mathcal{H}}^k(E)=0$. We show this exponents’ condition to be sharp by constructing, for any $\alpha <1-(2-\frac{1}{q})s$, a surface $S$ in the same class as above and a set $E\subset \tau (S,V)$ with ${\mathbb{1}}_E\in W^{s,1}(S)$ 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.