On Rough Frobenius-type Theorems and Their Hölder Estimates

Abstract

The thesis studies Frobenius-type theorems in non-smooth settings. We extend the definition of involutivity to non-Lipschitz subbundles using generalized functions. We prove the real Frobenius Theorem with sharp regularity on log-Lipschitz subbundles. We also develop a singular version of the Frobenius theorem on log-Lipschitz vector fields: if $X_1,\dots,X_m$ are log-Lipschitz vector fields such that $[X_i,X_j]=\sum_{k=1}^mc_{ij}^kX_k$ where $c_{ij}^k$ are the derivatives of log-Lipschitz functions, then for any point $p$ there is a $C^1$-manifold containing $p$ such that $X_1,\dots,X_m$ span its tangent space. On the quantitative side, if $c_{ij}^k\in C^{α-1}$ where $1<α<2$ then on each leaf where $X_1,\dots,X_m$ span the tangent spaces we can find a regular parameterization $Φ$ such that $Φ^*X_1,\dots,Φ^*X_m$ are $C^α$, and their $C^α$ norm depend only on the diffeomorphic invariant quantities of $X_1,\dots,X_m$. For a complex Frobenius structure there is a coordinate chart $F$ that takes image in $\mathbb R^r_t\times\mathbb C^m_z\times \mathbb R^{N-r-2m}_s$, such that the structure is locally spanned by $F^*\partial_t,F^*\partial_z$. When it has Hölder regularity $α>1$, we show that the coordinate chart $F$ may be taken to be $\mathscr C^α$, and the vector fields $F^*\partial_t,F^*\partial_z$ are $\mathscr C^{α-ε}$ for every $ε>0$. We give an example to show that the regularity result for $F^*\partial_z$ is optimal. When a complex Frobenius structure $S$ is $C^α$ ($\frac12<α\le1$) such that $S+\bar S$ is log-Lipschitz, then for every $ε>0$ there is a $C^{2α-1-ε}$ homeomorphism $Φ(t,z,s)$ such that $S$ is spanned by $Φ_*\partial_t,Φ_*\partial_z\in C^{2α-1-ε}$.