BMO embeddings, chord-arc curves, and Riemann mapping parametrization
BMO embeddings, chord-arc curves, and Riemann mapping parametrization
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DOI：
10.1016/j.aim.2023.108933
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发表时间：
2021-11
影响因子：
1.7
通讯作者：
Huaying Wei;Katsuhiko Matsuzaki
中科院分区：
文献类型：
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作者：
Huaying Wei;Katsuhiko Matsuzaki
我们考虑平面上弦-弧曲线在实直线上的参数化γ空间，并将其嵌入到BMO Teichmüler空间的乘积中。我们证明的关于这种表示的基本定理是：γ↦LOGγ‘是到BMO函数的复Banach空间的双全纯同胚。利用这两个等价的复结构，我们清楚地阐述了某些子空间之间涉及的映射的解析相依性。特别研究了弦-弧曲线的黎曼映射及其对弧长参数的依赖关系。因此，我们可以通过证明这种依赖不是连续的来解决Katznelson、Nag和Sullivan的猜想。
We consider the space of chord-arc curves on the plane passing through infinity with their parametrization γ defined on the real line, and embed this space into the product of the BMO Teichmüller spaces. The fundamental theorem we prove about this representation is that γ↦ log γ′ is a biholomorphic homeomorphism into the complex Banach space of BMO functions. Using these two equivalent complex structures, we develop a clear exposition on the analytic dependence of involved mappings between certain subspaces. Especially, we examine the parametrization of a chord-arc curve by using the Riemann mapping and its dependence on the arc-length parametrization. As a consequence, we can solve the conjecture of Katznelson, Nag, and Sullivan by showing that this dependence is not continuous.