2019 Fiscal Year Annual Research Report
New perspectives on space-time estimates for dispersive equations
| Project/Area Number |
19H01796
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| Research Institution | Saitama University |
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Principal Investigator |
BEZ NEAL 埼玉大学, 理工学研究科, 教授 (30729843)
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| Co-Investigator(Kenkyū-buntansha) |
杉本 充 名古屋大学, 多元数理科学研究科, 教授 (60196756)
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| Project Period (FY) |
2019-04-01 – 2023-03-31
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| Keywords | Strichartz estimates / Orthonormal data / Maximal estimates / Oscillatory integrals |
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| Outline of Annual Research Achievements |
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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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
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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| Strategy for Future Research Activity |
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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