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2016 年度 実績報告書

Conjectures associated with Brascamp-Lieb type inequalities

研究課題

研究課題/領域番号 16H05995
研究機関埼玉大学

研究代表者

BEZ NEAL  埼玉大学, 研究機構研究企画推進室, 准教授 (30729843)

研究期間 (年度) 2016-04-01 – 2019-03-31
キーワードBrascamp-Lieb inequality / Stability
研究実績の概要

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.

現在までの達成度 (区分)
現在までの達成度 (区分)

1: 当初の計画以上に進展している

理由

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.

今後の研究の推進方策

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.

  • 研究成果

    (9件)

すべて 2017 2016

すべて 雑誌論文 (1件) (うち国際共著 1件、 査読あり 1件、 謝辞記載あり 1件) 学会発表 (7件) (うち国際学会 6件、 招待講演 7件) 学会・シンポジウム開催 (1件)

  • [雑誌論文] Behaviour of the Brascamp-Lieb constant2017

    • 著者名/発表者名
      Jonathan Bennett, Neal Bez, Michael Cowling, Taryn Flock
    • 雑誌名

      Bulletin of the London Mathematical Society

      巻: - ページ: -

    • DOI

      10.1112/blms.12049

    • 査読あり / 国際共著 / 謝辞記載あり
  • [学会発表] Stability of the Brascamp-Lieb inequality2017

    • 著者名/発表者名
      Neal Bez
    • 学会等名
      Analysis Seminar
    • 発表場所
      University of Edinburgh (United Kingdom)
    • 年月日
      2017-03-20 – 2017-03-20
    • 国際学会 / 招待講演
  • [学会発表] Smoothing estimates for the kinetic transport equation via the cone multiplier2017

    • 著者名/発表者名
      Neal Bez
    • 学会等名
      Harmonic Analysis Workshop
    • 発表場所
      金沢大学 (石川県金沢市)
    • 年月日
      2017-03-05 – 2017-03-05
    • 招待講演
  • [学会発表] Smoothing estimates for velocity averages via the cone multiplier2017

    • 著者名/発表者名
      Neal Bez
    • 学会等名
      Harmonic Analysis and Applications
    • 発表場所
      Seoul National University (South Korea)
    • 年月日
      2017-03-01 – 2017-03-01
    • 国際学会 / 招待講演
  • [学会発表] Stability of the Brascamp-Lieb constant and applications2016

    • 著者名/発表者名
      Neal Bez
    • 学会等名
      International Conference for the 70th Anniversary of the Korean Mathematical Society
    • 発表場所
      Seoul National University (South Korea)
    • 年月日
      2016-10-22 – 2016-10-22
    • 国際学会 / 招待講演
  • [学会発表] Estimates for the kinetic transport equation in hyperbolic Sobolev spaces2016

    • 著者名/発表者名
      Neal Bez
    • 学会等名
      Analysis Seminar
    • 発表場所
      Seoul National University (South Korea)
    • 年月日
      2016-10-19 – 2016-10-19
    • 国際学会 / 招待講演
  • [学会発表] Stability of the Brascamp-Lieb constant and applications2016

    • 著者名/発表者名
      Neal Bez
    • 学会等名
      4th East Asian Conference in Harmonic Analysis and Applications
    • 発表場所
      Yonsei University (South Korea)
    • 年月日
      2016-08-05 – 2016-08-05
    • 国際学会 / 招待講演
  • [学会発表] Strichartz estimates for the kinetic and Schrodinger equations2016

    • 著者名/発表者名
      Neal Bez
    • 学会等名
      Analysis Seminar
    • 発表場所
      Seoul National University (South Korea)
    • 年月日
      2016-04-29 – 2016-04-29
    • 国際学会 / 招待講演
  • [学会・シンポジウム開催] Interactions Between Harmonic And Geometric Analysis2016

    • 発表場所
      埼玉大学・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 (東京都千代田区)
    • 年月日
      2016-11-28 – 2016-12-01

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公開日: 2018-01-16  

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