Modulation spaces and HRT conjecture

• Project/Area Number: 19K03533
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12010:Basic analysis-related
• Research Institution: Hokkaido University
• Principal Investigator: Kobayashi Masaharu 北海道大学, 理学研究院, 准教授 (30516480)
• Project Period (FY): 2019-04-01 – 2022-03-31
• Project Status: Completed (Fiscal Year 2021)
• Budget Amount: ¥2,340,000 (Direct Cost: ¥1,800,000、Indirect Cost: ¥540,000); Fiscal Year 2021: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2020: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2019: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
• Keywords: モジュレーション空間 / 短時間フーリエ変換 / 分散型方程式 / フーリエ級数 / 作用関数 / フーリエ・ルベーグ空間 / 高階分散型方程式 / フーリエ係数 / 関数空間論 / HRT予想

Outline of Research at the Start

ホログラフィーの研究でノーベル物理学賞を受賞したD.Gabor氏が用いた「Gauss関数の平行移動と変調により生成される関数系を用いて、Fourier級数展開のように関数を展開する」というアイデアに起源をもち、これまで互いに影響し合いながら発展してきた研究テーマである「Modulation空間」と「HRT予想」を調和解析及び実解析的手法を用いて研究する。

Outline of Final Research Achievements

Throughout this project, we have studied the basic properties of function spaces and their applications to PDEs. Especially, we obtain the followings:(1) Using the short-time Fourier transform, we proposed a new representation of the solution of higher-order dispersive equations (including free Schrodinger equation and Airy equation) (2) We obtained Katznelson-type and Leblanc-type theorem concerning the characterization of operating functions in the function space consisting of functions whose Fourier coefficients belong to the weighted sequence space.

Academic Significance and Societal Importance of the Research Achievements

モジュレーション空間（やそれに関連するような関数空間）に関する研究はまだ日が浅く、「モジュレーション空間と相性のよい偏微分方程式は何か？」や「モジュレーション空間における作用関数を特徴づけられるか？」など多くの問題が存在する。今回得られた結果はこれらの問題の解決に重要な役割を果たすと考えられる。また、今回得られた成果は調和解析や偏微分方程式の研究において表される様々な関数空間や作用素の研究にも応用可能であると思われる。

Research Products

• [Journal Article] Operating functions on A^q_s (T) (2022)
  • Author(s): M. Kobayashi, E. Sato
  • Journal Title: Journal of Fourier Analysis and Applications Volume: －
  • Peer Reviewed
• [Journal Article] Representation of higher-order dispersive operators via short-time Fourier transform and its application (2021)
  • Author(s): Keiichi Kato, Masaharu Kobayashi, Shingo Ito, Tadashi Takahashi
  • Journal Title: Tohoku Math. J. (2) Volume: 73 (1) Issue: 1 Pages: 105-118
  • DOI: 10.2748/tmj.20191226
  • Peer Reviewed
• [Journal Article] REPRESENTATION OF HIGHER-ORDER DISPERSIVE OPERATORS VIA SHORT-TIME FOURIER TRANSFORM AND ITS APPLICATION (2020)
  • Author(s): K. Kato, M. Kobayashi, S. Ito and T. Takahashi
  • Journal Title: Tohoku Math. Journal Volume: -
  • Peer Reviewed
• [Presentation] Operating functions on A^q_s (T) (2021)
  • Author(s): M. Kobayashi
  • Organizer: the 13th ISAAC Congress
  • Int'l Joint Research / Invited
• [Presentation] Representation of higher-order dispersive operators via short-time Fourier transform and its application (2019)
  • Author(s): M. Kobayashi
  • Organizer: Function Spaces and Geometric Analysis and Their Applications
  • Int'l Joint Research / Invited
• [Presentation] Operating functions on A^q_s (T) (2019)
  • Author(s): M. Kobayashi
  • Organizer: 12th International ISAAC Congress
  • Int'l Joint Research / Invited