| 研究課題/領域番号 |
19H01796
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| 研究機関 | 埼玉大学 |
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研究代表者 |
BEZ NEAL 埼玉大学, 理工学研究科, 教授 (30729843)
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| 研究分担者 |
杉本 充 名古屋大学, 多元数理科学研究科, 教授 (60196756)
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| 研究期間 (年度) |
2019-04-01 – 2023-03-31
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| キーワード | Strichartz estimates / Orthonormal data / Maximal estimates / Oscillatory integrals |
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| 研究実績の概要 |
The primary focus of the research this year has been to develop a systematic theory of Strichartz estimates for orthonormal systems of initial data. This is one of the main goals of the original research proposal, and the ultimate aim is to obtain an abstract theory in the spirit of the work on classical Strichartz estimates by Keel and Tao. Substantial progress has been made in this direction, and new results have been obtained in the case of the wave equation, Klein-Gordon equation, and the fractional Schrodinger equations. As a related line of research which has naturally evolved during the course of this research project, we have initiated the study of the pointwise convergence problem associated with systems of infinitely many fermions. The single-particle problem is known as Carleson’s problem and has attracted significant attention since its formulation in the early 1980s. In this direction, we have obtained some sharp results in the one-dimensional version of the problem.
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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
2: おおむね順調に進展している
理由
We have obtained new results on Strichartz estimates for orthonormal systems associated with the wave, Klein-Gordon, and fractional Schrodinger equations. In order to accomplish this, we overcame a significant technical barrier present in earlier work of Frank-Sabin by establishing certain weighted oscillatory integral estimates. Our work also makes contact with a significant literature on damped oscillatory integral estimates and opens up a new line of research which seeks to make a unification of the known estimates in a natural geometric framework. In an independent paper on Carleson’s problem for infinitely many fermions, we simultaneously address an endpoint problem of Frank-Sabin regarding Strichartz estimates for the Schrodinger equation for orthonormal systems of data.
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| 今後の研究の推進方策 |
The next phase of the project will focus on developing further the theory of space-time estimates associated with orthonormal systems of initial data. This will include extending the results we have already obtained regarding Strichartz estimates and, in addition, developing the theory of so-called Kato smoothing estimates associated with orthonormal systems.
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