Conjectures associated with Brascamp-Lieb type inequalities

2018 Fiscal Year Final Research Report

• Project/Area Number: 16H05995
• Research Category: Grant-in-Aid for Young Scientists (A)
• Allocation Type: Single-year Grants
• Research Field: Basic analysis
• Research Institution: Saitama University
• Principal Investigator: BEZ Neal 埼玉大学, 理工学研究科, 准教授 (30729843)
• Research Collaborator: BENNETT Jonathan ; COWLING Michael ; CUNANAN Jayson ; FLOCK Taryn ; GUTIERREZ Susana ; ILIOPOULOU Marina ; JEAVONS Chris ; LEE Sanghyuk ; NAKAMURA Shohei ; OZAWA Tohru ; SAWANO Yoshihiro ; SUGIMOTO Mitsuru
• Project Period (FY): 2016-04-01 – 2019-03-31
• Keywords: Brascamp-Lieb不等式 / 安定性

Outline of Final Research Achievements

A key outcome of this research project was the successful complete verification of the nonlinear Brascamp-Lieb (BL) conjecture. This was one of the main goals of the initial research proposal and has been achieved through a series of papers. The first step was to establish local boundedness of the BL constant in the standard version of the BL inequality and use this to prove the nonlinear BL conjecture for input functions with arbitrarily small Sobolev regularity. As a further application of this stability result, we advanced the theory of the multilinear Fourier restriction problem. Next, we showed continuity of the BL constant and this result was used in our recent complete resolution of the nonlinear BL conjecture. Additionally, this research project has produced various new developments on related theory, including estimates for the kinetic transport equation.

Free Research Field

調和解析

Academic Significance and Societal Importance of the Research Achievements

非線形ブラスカンプ・リーブ予想を証明した事により、抽象調和解析や偏微分方程式への応用を行った。例えば、リー群に関する局所的なヤングの畳み込み不等式に対する最良定数を求めた。更に、ザハロフ方程式系の理論における幾何的な不等式を示す事にも成功した。このように本研究の成果は、将来的にも多分野で応用される事が期待される。

調和解析および幾何解析の研究交流を更に推進するために、4日間の国際的な研究集会を東京ステーションカレッジ・埼玉大学サテライトキャンパスで開催した。