Theory of the universal Teichmüller space in harmonic analysis
| Project/Area Number |
21F20027
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| Research Category |
Grant-in-Aid for JSPS Fellows
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| Allocation Type | Single-year Grants |
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| Section | 外国 |
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| Review Section |
Basic Section 12010:Basic analysis-related
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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
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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| Project Status |
Completed (Fiscal Year 2022)
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| Budget Amount *help |
¥2,300,000 (Direct Cost: ¥2,300,000)
Fiscal Year 2022: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2021: ¥1,200,000 (Direct Cost: ¥1,200,000)
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| Keywords | 複素解析学 |
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| Outline of Research at the Start |
普遍タイヒミュラー空間の理論は，数学や数理物理学において無限次元の自由度をもつ対象のパラメーター空間として有用である．対象の性質によりタイヒミュラー空間に属する写像族に制限を与え，各種の部分空間が定義できる．本研究では，その理論の普遍的な枠組みを与えることを目標とし，とくに調和解析的な理論が適用可能な BMO 関数を中心として，そこから派生する種々のタイヒミュラー空間の解析を行う．
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| Outline of Annual Research Achievements |
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.
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| Research Progress Status |
令和4年度が最終年度であるため、記入しない。
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| Strategy for Future Research Activity |
令和4年度が最終年度であるため、記入しない。
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Report
(2 results)
Research Products
(16 results)
