2021 Fiscal Year Annual Research Report
Theory of the universal Teichmüller space in harmonic analysis
| Project/Area Number |
21F20027
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| Research Institution | Waseda University |
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Principal Investigator |
松崎 克彦 早稲田大学, 教育・総合科学学術院, 教授 (80222298)
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| Co-Investigator(Kenkyū-buntansha) |
Wei Huaying 早稲田大学, 教育・総合科学学術院, 外国人特別研究員
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| Project Period (FY) |
2021-04-28 – 2023-03-31
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| Keywords | 複素解析学 |
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| Outline of Annual Research Achievements |
In our research, we focus on subspaces of the universal Teichmueller space corresponding to locally rectifiable quasicircles, which have already been extensively studied individually. By distilling the essence from each argument and combining novel tools and methods with existing ones, we have obtained a series of results that have received high acclaim among our peers. The specific findings are outlined below.
(1) We provide a real-analytic section for the Teichmueller projection onto the VMO-Teichmueller space, utilizing a variant of the Beurling-Ahlfors extension by heat kernel introduced by Fefferman et al. (Ann Math 134: 65-124, 1991). Building on this result, we establish that the VMO-Teichmueller space can be endowed with a real Banach manifold structure, which is analytically equivalent to its complex Banach manifold structure. Additionally, we demonstrate the existence of a real-analytic contraction mapping in the VMO-Teichmueller space.
(2) We investigate the biholomorphic correspondence between the space of p-Weil-Petersson curves γ on the plane and the p-Besov space u=log γ' on the real line, where p>1. We apply a variant of the Beurling-Ahlfors extension, employing the heat kernel, to extend quasiconformal curves. From these analyses, we derive several immediate consequences that elucidate the analytic structures pertaining to the parameter spaces of p-Weil-Petersson curves, and establish fundamental properties of the p-Weil-Petersson Teichmueller space.
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| Current Status of Research Progress |
Current Status of Research Progress
1: Research has progressed more than it was originally planned.
Reason
We have succeeded in constructing various discussions to enhance the understanding of previous results, and it has become evident that this approach is highly effective. We have restructured the previous research for better clarity, and in addition, new achievements have been obtained one after another. The collaborative research framework has also proven to be effective, resulting in an extremely productive research environment.
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| Strategy for Future Research Activity |
We conduct research on the Teichmueller theory of chord-arc curves and Weil-Petersson curves, employing quasiconformal analysis as our primary approach and utilizing Muckenhoupt theory as a tool. However, despite the widespread use of these methods, they alone are insufficient to tackle the long-standing open problems in this field. In order to study problems related to plane curves from a harmonic analysis perspective, we adopt a different approach. Rather than solely focusing on the curve itself, we consider the curve together with its parametrization. By incorporating the complex structure derived from Bers theory of simultaneous uniformization onto the space of parametrized curves, we gain a clear and straightforward understanding of the curve's structure. This novel method, which has been overlooked in existing literature, enables us to simplify certain established results and explore various open problems with greater depth.
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