Frobenius theorem and fine structure of tangency sets to non-involutive distributions
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2025
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| author | Alberti, Giovanni Massaccesi, Annalisa Merlo, Andrea |
| author_facet | Alberti, Giovanni Massaccesi, Annalisa Merlo, Andrea |
| contents | 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 $τ(S,V)$ 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 $W^{α,q}$ and if a Borel set $E \subset τ(S,V)$ satisfies $ \mathbb1_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 $\mathscr{H}^k(E) = 0$. We show this exponents' condition to be sharp by constructing, for any $α< 1 - \bigl(2 - \tfrac{1}{q}\bigr) s$, a surface $S $ in the same class as above and a set $E \subset τ(S,V)$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_03715 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Frobenius theorem and fine structure of tangency sets to non-involutive distributions Alberti, Giovanni Massaccesi, Annalisa Merlo, Andrea Differential Geometry Analysis of PDEs Classical Analysis and ODEs 58A30, 53C17, 58A25, 35R03 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 $τ(S,V)$ 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 $W^{α,q}$ and if a Borel set $E \subset τ(S,V)$ satisfies $ \mathbb1_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 $\mathscr{H}^k(E) = 0$. We show this exponents' condition to be sharp by constructing, for any $α< 1 - \bigl(2 - \tfrac{1}{q}\bigr) s$, a surface $S $ in the same class as above and a set $E \subset τ(S,V)$ 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. |
| title | Frobenius theorem and fine structure of tangency sets to non-involutive distributions |
| topic | Differential Geometry Analysis of PDEs Classical Analysis and ODEs 58A30, 53C17, 58A25, 35R03 |
| url | https://arxiv.org/abs/2506.03715 |