2018 Fiscal Year Annual Research Report
New developments on the restriction conjecture for the Fourier transform using multilinear analysis
| Project/Area Number |
18F18020
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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, 准教授 (30729843)
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| Co-Investigator(Kenkyū-buntansha) |
CUNANAN JAYSON MESITAS 埼玉大学, 理工学研究科, 外国人特別研究員
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| Project Period (FY) |
2018-04-25 – 2020-03-31
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| Keywords | Strichartz estimates / Velocity average / Smoothing estimates |
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| Outline of Annual Research Achievements |
New results concerning inhomogeneous Strichartz estimates for the wave equation have been obtained. After establishing that such estimates fail to be true outside the so-called acceptable region, we consider inhomogeneous Strichartz estimates on a critical line where the acceptability condition marginally fails. In such a critical case, we establish weak-type estimates. In order to achieve this, we prove that such weak-type estimates hold in an abstract setting and deduce the special case of the wave equation via an application of the dispersive estimate for the wave equation in certain Besov spaces.
In a different direction, new results have been obtained concerning smoothing estimates for velocity averages of solutions to the kinetic transport equation. In particular, in the setting of square-integrable functions, we have established a definitive result concerning smoothing estimates for velocity averages measured in hyperbolic Sobolev spaces, where the initial data is radial and where the velocity domain is either a sphere or a ball. We have also obtained extensions of these estimates to a mixed-norm setting. Our arguments rely on a variety of tools from harmonic analysis, including restriction theory of the Fourier transform.
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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
The original proposal centred on the restriction theory of the Fourier transform. Such theory has very close connections with Strichartz estimates and our progress in the direction of inhomogeneous Strichartz estimates for the wave equation is an important contribution to this topic. In addition, we have made progress on other topics by making use of restriction theory, including smoothing estimates for velocity averages for solutions of the kinetic transport equation.
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| Strategy for Future Research Activity |
The goal for the next year of this project will focus on Strichartz estimates and related inequalities. For example, our previous work concerning velocity averages for the kinetic transport equation naturally gives motivation to study certain mixed-norm estimates for Fourier multiplier operators which are associated with hypersurfaces. It is anticipated that the geometry of the underlying hypersurface will play a pivotal role.
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Research Products
(6 results)
