2023 Fiscal Year Annual Research Report
Investigating the stability of the inverse Brascamp-Lieb inequality
| Project/Area Number |
23H01080
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| Allocation Type | Single-year Grants |
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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) |
中村 昌平 大阪大学, 大学院理学研究科, 助教 (30896121)
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| Project Period (FY) |
2023-04-01 – 2027-03-31
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| Keywords | Best constant / Inverse Brascamp-Lieb / Regularity / Minimizers |
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| Outline of Annual Research Achievements |
The inverse Brascamp-Lieb inequality is similar to the classical (forward) Brascamp-Lieb inequality but the direction of the inequality is reversed. Although key results such as gaussian saturation and a characterisation of feasibility have already been established in prior work for the inverse Brascamp-Lieb inequality, a number of fundamental problems remain open and the main focus of the research this year has been understanding the regularity of the optimal constant in the inverse Brascamp-Lieb inequality with respect to the underlying linear transformations. Partial progress has been made on this problem in the sense that continuity of the inverse Brascamp-Lieb constant has been established for certain classes of linear transformations. In related work, again for certain classes of linear transformations, progress has also been made on obtaining a characterisation of minimizing input functions.
Progress has also been made in the direction of applications of multilinear analysis to the mathematical theory of dispersive partial differential equations, as well as stability of functional inequalities related to the Brascamp-Lieb inequality such as Nelson's hypercontractivity inequality and the logarithmic Sobolev inequality.
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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
Many aspects of the theory of the classical (forward) Brascamp-Lieb inequality are by now very well developed, and this includes many results in the direction of understanding the regularity of the best constant, and characterising which functions attain the best constant, etc. Although it seems reasonable that one should be able to extend such results to the context of the inverse Brascamp-Lieb inequality, there seem to be significant difficulties in doing so. Despite this, partial progress has been made in this research project and the desired results appear to be achievable in the near future.
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| Strategy for Future Research Activity |
The next phase of the project will focus on extending the results that have already been obtained with regard to the regularity of the inverse Brascamp-Lieb constant and characterisation of minimisers.
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