2017 Fiscal Year Annual Research Report
Conjectures associated with Brascamp-Lieb type inequalities
| Project/Area Number |
16H05995
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| Research Institution | Saitama University |
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Principal Investigator |
BEZ NEAL 埼玉大学, 研究機構, 准教授 (30729843)
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| Project Period (FY) |
2016-04-01 – 2019-03-31
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| Keywords | Brascamp-Lieb inequality / Schrodinger equation / Transport equation / Trace estimates |
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| Outline of Annual Research Achievements |
Progress has been made on understanding the Brascamp-Lieb inequality in the framework of Lorentz spaces. In particular, in joint work with Sanghyuk Lee, Shohei Nakamura and Yoshihiro Sawano, necessary conditions have been obtained which establish the sharpness of the range of possible Lorentz exponents in the so-called subcritical case. Sharpness of some related estimates for the kinetic transport equation has also been obtained by similar ideas. Continuing in this direction, in collaboration with Jayson Cunanan and Sanghyuk Lee, harmonic analysis techniques were used to establish smoothing estimates for the kinetic transport equation at a certain critical regularity, thus solving an open problem left open in earlier work on this research project.
Further results obtained on this research project this year include some novel space-time estimates for the Schrodinger equation involving the X-ray transform; this is joint work with Jonathan Bennett, Taryn Flock, Susana Gutierrez and Marina Iliopoulou. In a further joint work with Jonathan Bennett, we develop a new perspective on the so-called heat-flow monotonicity method in the context of geometric inequalities, including the Brascamp-Lieb inequality and hypercontractivity. Also in the context of geometric inequalities, in joint work with Chris Jeavons, Tohru Ozawa and Mitsuru Sugimoto, new results on the trace theorem on the sphere have been obtained.
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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
Substantial progress has been obtained towards the main goal of the original proposal of this research project, the nonlinear Brascamp-Lieb conjecture, and it is anticipated that this part of the project will reach completion soon. Furthermore, unexpectedly good progress has been made on new lines of research inspired by the original research goals. This includes new perspectives on delicate boundedness properties of the solutions to the Schrodinger and the kinetic transport equations. Developments have also been made on geometric estimates, such as the fundamental trace estimate on the sphere, where a new understanding of the stability of the estimate and properties of its near-extremisers has been reached.
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| Strategy for Future Research Activity |
The main purpose is to establish the nonlinear Brascamp-Lieb conjecture in full generality. In some work in progress, partial breakthroughs have already been made in the so-called subcritical case. A major goal of this research project is to establish the conjecture in all cases and thus complete the essential part of the first programme of the original research application. Once the conjecture has been verified, efforts will be made to apply the theory in various directions, including PDE and euclidean harmonic analysis.
In a different line of development, it is also planned to utilise harmonic analysis techniques to make novel contributions to the recently emerging theory of Strichartz estimates for orthonormal systems of data.
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Research Products
(9 results)
