Theory of the universal Teichmüller space in harmonic analysis

普遍的泰希穆勒空间理论由于其与数学其他分支的密切联系而非常活跃。在我们的研究中，我们研究的Teichmueller空间是通过将调和分析的一定程度的正则性纳入拟圆而得到的。具体来说，我们关注与弦弧曲线、渐近光滑曲线和Weil-Petersson曲线相关的Teichmueller空间。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曲线则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)我们研究了平面上的弦弧曲线空间，并在实线上定义了弦弧曲线的参数化。我们将这个空间嵌入到BMO Teichmueller空间的产品中。通过沿着这条线展开论证，我们能够简化Coifman和Meyer的定理，并且我们可以为Katznelson-Nag-Sullivan提出的问题提供一个否定的答案。(2)利用弦Loewner理论，我们推广了准共形扩展的Ahlfors-Weill公式，并在20世纪80年代Becker在磁盘上的工作的基础上，建立了半平面上该结果的一个版本。作为这个拟共形扩展的一个应用，我们利用由Schwarzian导数引起的消失Carleson测量条件在半平面上刻画了VMO-Teichmueller空间的一个元素。

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.