论文标题
在粗糙的Frobenius型定理上及其Hölder估计
On Rough Frobenius-type Theorems and Their Hölder Estimates
论文作者
论文摘要
论文研究非平滑环境中的frobenius型定理。 我们使用广义函数扩展了对非lipschitz子捆绑的定义。我们在log-lipschitz子捆绑中证明了真正的Frobenius定理。我们还在log-lipschitz向量字段上开发了frobenius定理的单数版本：如果$ x_1，\ dots，x_m $是log-lipschitz vector字段，以至于$ [x_i，x_j] = \ sum_ = \ sum_ {k = 1}^mc_ mc_}^mc_}函数，然后在任何点$ p $中都有一个$ c^1 $ - manifold，其中包含$ p $，因此$ x_1，\ dots，x_m $跨越了其切线空间。在定量方面，如果$ c_ {ij}^k \在c^{α-1} $中，其中$ 1 <α<2 $，则在每个叶子上，其中$ x_1，\ dots，x_m $ span span the Pintent the Pintent Space我们可以找到一个常规的参数化$φ$φ$标准仅取决于$ x_1，\ dots，x_m $的差异不变量。 对于复杂的Frobenius结构，有一个坐标图$ F $在$ \ Mathbb r^r_t \ times \ times \ Mathbb C^M_z \ times \ Mathbb r^{n-r-2m} _s $中，因此该结构由$ f^*\ partial_t，f^*p partial partial partial partial partial partial f^*\ partial_t $。当它具有Hölder规律性$α> 1 $时，我们表明坐标表$ f $可能被认为为$ \ Mathscr c^α$，而vector fields $ f^*\ partial_t，f^*\ partial_z $ as $ \ sathscr c^{α-α-ε} $ for avery $ $ε> 0 $ 0。我们举例说明$ f^*\ partial_z $的规律性结果是最佳的。当复杂的Frobenius结构$ s $是$ c^α$（$ \ frac12 <α\ le1 $）时，$ s+s+s+bar s $是log-lipschitz，那么，对于每个$ε> 0 $，就有$ c^{2α-1-} $ somomorphism $ somomorphism $（t，t，z）$是$ s $ SY $φ_*\ partial_t，φ_*\ partial_z \ in C^{2α-1-ε} $。
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 $Φ_*\partial_t,Φ_*\partial_z\in C^{2α-1-ε}$.
