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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.

普适Teichmueller空间理论由于与数学的其他分支有着密切的联系而非常活跃。在我们的研究中，我们调查的Teichmueller空间是通过将一定程度的正则性从调和分析到拟圆中得到的。具体来说，我们专注于Teichmueller空间与弦弧曲线，渐近光滑曲线，和Weil-Petersson曲线。弦-弧曲线是谐波分析中的一个重要研究课题，而渐近光滑曲线和Weil-Petersson曲线都属于弦-弧曲线的范畴。Weil-Petersson曲线的研究是由SLE理论推动的。在我们的研究中，我们得到了以下关于弦弧曲线空间的结果：（1）研究了通过无穷远平面上的弦弧曲线空间，它们的参数化定义在真实的直线上。我们把这个空间嵌入到BMO Teichmueller空间的乘积中。通过沿着这条路线展开论证，我们能够简化Coifman和Meyer的一个定理，并且我们能够对Katznelson-Nag-Sullivan提出的一个问题给出否定的答案。(2)利用弦Loewner理论，我们推广了Ahlfors-Weill公式的拟共形扩张，并建立了一个版本的半平面，贝克尔的工作在20世纪80年代的磁盘上。作为这种拟共形推广的应用，我们利用Schwarzian导数诱导的Carleson测度消失条件刻画了半平面上VMO-Teichmueller空间的一个元素.