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

TL;DR

This paper investigates the extension of Frobenius' Theorem to surfaces with regularity below $C^{1,1}$, analyzing the structure of tangency sets to non-involutive distributions and establishing sharp conditions for their measure-theoretic properties.

Contribution

It provides a Frobenius-type result for low-regularity surfaces and characterizes the measure of tangency sets using fractional Sobolev spaces, with sharpness results.

Findings

If a set's characteristic function is in W^{s,1} with s>1/2, the set is null in the surface.

Under certain regularity conditions, tangency sets have zero Hausdorff measure.

Sharpness of exponents is demonstrated through explicit constructions.

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 $\tau(S,V)$ of $S$ to $V$ and $\mathbb1_E\in W^{s,1}(S)$ with $s>1/2$ then $E$ must be $\mathscr{H}^k$-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 $ \mathbb1_E\in W^{s,1}(S)$ with \[ s \in \bigl(0,\tfrac{1}{2}\bigr]\qquad\text{and}\qquad\alpha \;>\; 1 - \Bigl(2 - \tfrac{1}{q}\Bigr) \, s, \] then $\mathscr{H}^k(E) = 0$. We show this exponents' condition to be sharp by constructing, for any $\alpha < 1 - \bigl(2 - \tfrac{1}{q}\bigr) s$, a surface $S $ in the same class as above and a set $E \subset \tau(S,V)$ with $\mathbb1_E \in W^{s,1}(S)$ and $\mathscr{H}^k(E) > 0$. 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.