2022/02/15 by Liding Yao, Yao, Liding
Mathematics · #32Q60 #53C15 and 53C12 (Secondary) #58A30 (Primary) #Advanced Differential Equations and Dynamical Systems #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2202.07729
openalex publication_date 2022/02/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Rr×\mathbb Cm× \mathbb RN-r-2m through a coordinate chart F in such a way that the structure is locally spanned by F^*\frac∂∂ t1,…,F^*\frac∂∂ tr,F^*\frac∂∂ z1,…,F^*\frac∂∂ zm, where we have given \mathbb Rr×\mathbb Cm ×\mathbb RN-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 Rr×\mathbb Cm ×\mathbb RN-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∂∂ t1,…,F^*\frac∂∂ tr,F^*\frac∂∂ z1,…,F^*\frac∂∂ zm on the original manifold have Hölder-Zygmund regularity of order α-ε for every ε>0, and we give an example to show that the regularity for F^*\frac∂∂ z is optimal.