Preprint

Inserted: 4 jun 2025
Last Updated: 5 jun 2025

Year: 2025

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 C¹ and a non-involutive k-distribution V , if E is a Borel set contained in the tangency set τ (S, V ) of S to V and 1_E ∈ W^{s,1}(S) with s > 12 then E must be H^k -null in S. In addition, if S is locally a graph of a C¹ function with gradient in W^{α,q} and if a Borel set E ⊂ τ (S, V ) satisfies 1_E ∈ W^{s,1}(S) with s ∈ (0, 12] and α > 1 −(2 − 1q)s, then H^k (E) = 0. We show this exponents’ condition to be sharp by constructing, for any α < 1 − (2 − 1q)s, a surface S in the same class as above and a set E ⊂ τ (S, V ) with 1_E ∈ W^{s,1}(S) and 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.

Keywords: non-involutive distributions, Frobenius theorem, Lusin’s theorem for gradients