2016 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, 研究機構研究企画推進室, 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 / Stability |
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| Outline of Annual Research Achievements |
The research output so far has focused on the stability of the Brascamp-Lieb inequality. In collaboration with Jonathan Bennett (University of Birmingham), Taryn Flock (University of Birmingham) and Sanghyuk Lee (Seoul National University), we completely solved the nonlinear Brascamp-Lieb conjecture for input functions with arbitrarily small Sobolev regularity; answering this conjecture was pin-pointed as one of the aims of Programme 1 of this research project.
We proved this conjecture by first establishing that the constant in the classical version of the Brascamp-Lieb inequality is locally bounded with respect to the underlying linear mappings. In the same paper, further applications were given, including far-reaching generalisations of the multilinear restriction and Kakeya theorems of Bennett-Carbery-Tao. These results have already found exciting applications in number theory, in particular, work of Bourgain-Demeter-Guth in their complete solution of Vinogradov’s mean value conjecture.
In a follow-up paper, in collaboration with Jonathan Bennett (University of Birmingham), Michael Cowling (University of New South Wales) and Taryn Flock (University of Birmingham), we strengthened the aforementioned stability result by showing that the constant in the classical version of the Brascamp-Lieb inequality depends continuously, but not always smoothly, on the underlying linear mappings.
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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
In addition to essentially completing resolving one of the specified aims for this year of the project (nonlinear Brascamp-Lieb conjecture), significant breakthroughs have also been made in the theory of the kinetic transport equation, and in particular regarding the regularity properties of the velocity average of the solution of this equation. In joint work with Jonathan Bennett (University of Birmingham), Susana Gutierrez (University of Birmingham) and Sanghyuk Lee (Seoul National University), we utilised the connection that the kinetic transport equation enjoys with special cases of the Brascamp-Lieb inequality (such as the Loomis-Whitney inequality) and Kakeya-type estimates, and based on techniques from related problems in harmonic analysis, we have fully developed the smoothing estimates for the velocity average in the naturally associated mixed-norm Bourgain-spaces.
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| Strategy for Future Research Activity |
In the next stage of this project, one of the targets is to address the specified aim in the proposal regarding applications to inverse problems and dispersive PDE.
In the former case, one of the first steps will be to investigate extensions of the classical Brascamp-Lieb inequality where the input functions are allowed to belong to Lorentz spaces (particularly relevant to such applications are the weak Lebesgue spaces).
In the latter case regarding dispersive PDE, it is planned to use harmonic analysis techniques to significantly advance the fundamental theory of the Schrodinger equation, with an emphasis on applications of the multilinear theory tightly connected to the Brascamp-Lieb inequality.
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