Theory of the universal Teichmüller space in harmonic analysis

普遍理论

基本信息

批准号：
21F20027
负责人：
松崎克彦
金额：
$ 1.47万
依托单位：
Waseda University
依托单位国家：
日本
项目类别：
Grant-in-Aid for JSPS Fellows
财政年份：
2021
资助国家：
日本
起止时间：
2021-04-28 至 2023-03-31
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-21F20027/
关键词：
複素解析学

项目摘要

The theory of the universal Teichmueller space is highly active due to its close connections with other branches of mathematics. In our study, the Teichmueller spaces we investigate are obtained by incorporating a certain level of regularity from harmonic analysis into quasicircles. Specifically, we focus on Teichmueller spaces associated with chord-arc curves, asymptotically smooth curves, and Weil-Petersson curves. Chord-arc curves are a prominent subject of research in harmonic analysis, while asymptotically smooth curves and Weil-Petersson curves fall under the category of chord-arc curves. The study of Weil-Petersson curves is motivated by SLE theory. In our research, we have obtained the following results concerning the space of chord-arc curves:(1) We examine the space of chord-arc curves on the plane that pass through infinity, with their parametrizations defined on the real line. We embed this space into the product of the BMO Teichmueller spaces. By developing the argument along this line, we are able to simplify a theorem by Coifman and Meyer, and we can provide a negative answer to a question raised by Katznelson-Nag-Sullivan.(2) Utilizing chordal Loewner theory, we generalize the Ahlfors-Weill formula for quasiconformal extension and establish a version of this result for the half-plane, building upon Becker's work in the 1980s on the disk. As an application of this quasiconformal extension, we characterize an element of the VMO-Teichmueller space on the half-plane by employing the vanishing Carleson measure condition induced by the Schwarzian derivative.

通用Teichmueller空间的理论由于与其他数学分支的紧密联系而高度活跃。在我们的研究中，我们研究的Teichmueller空间是通过将谐波分析的一定规律性纳入准圆的一定水平来获得的。具体而言，我们专注于与和弦曲线，渐近平滑曲线和Weil-Petersson曲线相关的Teichmueller空间。和弦曲线是谐波分析中研究的重要主题，而渐近平滑的曲线和Weil-Petersson曲线属于Chord-Arc曲线类别。 Weil-Petersson曲线的研究是由SLE理论激励的。在我们的研究中，我们获得了有关和弦曲线空间的以下结果：（1）我们检查了通过无穷大的平面上和弦曲线的空间，其参数定义在实线上。我们将这个空间嵌入了BMO Teichmueller空间的产品中。通过沿着这条线发展论点，我们能够简化Coifman and Meyer的定理，我们可以为Katznelson-Nag-Sullivan提出的问题提供负面答案。（2）利用和弦loewner理论，我们概括了Ahlfors - 用于准文献扩展的韦尔公式，并为半平面建立此结果的版本，这是贝克尔在1980年代在磁盘上的作品建立的。作为此准形式扩展的应用，我们通过采用由Schwarzian衍生产品引起的消失的Carleson测量条件来表征半平面上VMO-Teichmueller空间的元素。