Theory of the universal Teichmüller space in harmonic analysis

点击翻译按钮获取中文摘要

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.