论文标题
Nirenberg的复杂Frobenius Theorem的SharpHölder规律性
Sharp Hölder Regularity for Nirenberg's Complex Frobenius Theorem
论文作者
论文摘要
Nirenberg的著名复杂的Frobenius Theorem在本地结构上提供了必要和足够的条件，以便何时将多种形式归因于$ \ \ \ \ \ \ \ \ \ \ \ m \ timebb r^r \ times \ times \ mathbb c^m \ times \ times \ times \ times \ times \ times \ mathbb r^{n-r-2m} $ f^*\ frac \ partial {\ partial t^1}，\ dots，f^*\ frac \ partial {\ partial t^r}，f^*\ frac \ partial {\ partial {\ partial z^1}，\ dots，\ dots，\ dots，\ dots，f^*\ frac \ frac \ frac \ partial z^m} r^r \ times \ mathbb c^m \ times \ times \ mathbb r^{n-r-2m} $ coordinates $（t，z，s）$。在本文中，我们为实现这一实现的坐标图提供了最佳的Hölder-Zygmund规律性。也就是说，如果该结构具有hölder-zygmund的规律性$α> 1 $，则可以将其映射到$ \ MATHBB r^r \ times \ times \ times \ Mathbb c^m \ times \ Mathbb r^{n-r-2m} $的坐标图$ f $可以被视为具有Hölder-Zygmund undordity $ and order and Order and Orderive and and Orderive and and is and rescripity。此外，我们可以以某种方式选择此$ f $ z^1},\dots,F^*\frac\partial{\partial z^m}$ on the original manifold have Hölder-Zygmund regularity of order $α-\varepsilon$ for every $\varepsilon>0$, and we give an example to show that the regularity for $F^*\frac\partial{\partial z}$ is optimal.
Nirenberg's famous complex Frobenius theorem gives necessary and sufficient conditions on a locally integrable structure for when the manifold is locally diffeomorphic to $\mathbb R^r\times\mathbb C^m\times \mathbb R^{N-r-2m}$ through a coordinate chart $F$ in such a way that the structure is locally spanned by $F^*\frac\partial{\partial t^1},\dots,F^*\frac\partial{\partial t^r},F^*\frac\partial{\partial z^1},\dots,F^*\frac\partial{\partial z^m}$, where we have given $\mathbb R^r\times\mathbb C^m \times\mathbb R^{N-r-2m}$ coordinates $(t,z,s)$. In this paper, we give the optimal Hölder-Zygmund regularity for the coordinate charts which achieve this realization. Namely, if the structure has Hölder-Zygmund regularity of order $α>1$, then the coordinate chart $F$ that maps to $\mathbb R^r\times\mathbb C^m \times\mathbb R^{N-r-2m}$ may be taken to have Hölder-Zygmund regularity of order $α$, and this is sharp. Furthermore, we can choose this $F$ in such a way that the vector fields $F^*\frac\partial{\partial t^1},\dots,F^*\frac\partial{\partial t^r},F^*\frac\partial{\partial z^1},\dots,F^*\frac\partial{\partial z^m}$ on the original manifold have Hölder-Zygmund regularity of order $α-\varepsilon$ for every $\varepsilon>0$, and we give an example to show that the regularity for $F^*\frac\partial{\partial z}$ is optimal.
