On Rough Frobenius-type Theorems and Their H\"older 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 $c_{ij}^k\in C^{\alpha-1}$ where $1<\alpha<2$ then on each leaf where $X_1,\dots,X_m$ span the tangent spaces we can find a regular parameterization $\Phi$ such that $\Phi^*X_1,\dots,\Phi^*X_m$ are $C^\alpha$, and their $C^\alpha$ 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 H\"older regularity $\alpha>1$, we show that the coordinate chart $F$ may be taken to be $\mathscr C^\alpha$, and the vector fields $F^*\partial_t,F^*\partial_z$ are $\mathscr C^{\alpha-\epsilon}$ for every $\epsilon>0$. We give an example to show that the regularity result for $F^*\partial_z$ is optimal. When a complex Frobenius structure $S$ is $C^\alpha$ ($\frac12<\alpha\le1$) such that $S+\bar S$ is log-Lipschitz, then for every $\epsilon>0$ there is a $C^{2\alpha-1-\epsilon}$ homeomorphism $\Phi(t,z,s)$ such that $S$ is spanned by $\Phi_*\partial_t,\Phi_*\partial_z\in C^{2\alpha-1-\epsilon}$.