Multilinear harmonic analysis and the singularity

• Project/Area Number: 16K05201
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Osaka University
• Principal Investigator: Tomita Naohito 大阪大学, 理学研究科, 准教授 (10437337)
• Project Period (FY): 2016-04-01 – 2020-03-31
• Project Status: Completed (Fiscal Year 2019)
• Budget Amount: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2019: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2016: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: 多重線形作用素 / 擬微分作用素 / フーリエ乗法作用素 / フーリエマルチプライヤー

Outline of Final Research Achievements

In the field of harmonic analysis, the research to extend the theory for linear operators to the one for multilinear operators has been actively studied since around 2000. Nowadays this topic is often called multilinear harmonic analysis. Multilinear harmonic analysis is not just a generalization of linear theory, it is also a challenging problem for harmonic analysis, and it has the potential for the development of the study of partial differential equations from an application perspective. I studied how to approach operators with strong singularity such as bilinear Hilbert transform. I also obtained the results on the boundedness of bilinear pseudo-differential operators.

Academic Significance and Societal Importance of the Research Achievements

宮地晶彦氏(東京女子大学)との双線形擬微分作用素に関する研究は，V. Naibo 氏と A. Thomsom 氏の J. Math. Anal. Appl. (2019) の論文の中で，基本的な枠組みにおける有界性問題を終わらせたと紹介されており，価値あるものと信じている．また，L. Grafakos 氏，宮地晶彦氏との多重線形フーリエ乗法作用素の共同研究では，有界性を保証するためのマルチプライヤーに課すべき最適な正則性条件を決定することに成功した．

Research Products

• [Journal Article] Nonlinear operations on a class of modulation spaces (2020)
  • Author(s): Kato Tomoya、Sugimoto Mitsuru、Tomita Naohito
  • Journal Title: Journal of Functional Analysis Volume: 278 Issue: 9 Pages: 108447-108447
  • DOI: 10.1016/j.jfa.2019.108447
  • Peer Reviewed
• [Journal Article] Pseudodifferential operators with symbols in the Hormander class S^0_{\alpha,\alpha} on \alpha-modulation spaces (2019)
  • Author(s): T. Kato, N. Tomita
  • Journal Title: Monatshefte fur Mathematik Volume: 188 Issue: 4 Pages: 667-687
  • DOI: 10.1007/s00605-019-01264-y
  • Peer Reviewed
• [Journal Article] A remark on the Schrodinger propagator on Wiener amalgam spaces (2018)
  • Author(s): T. Kato, N. Tomita
  • Journal Title: Mathematische Nachrichten Volume: 印刷中 Issue: 2 Pages: 350-357
  • DOI: 10.1002/mana.201700445
  • Peer Reviewed
• [Journal Article] Weighted norm inequalities for multilinear Fourier multipliers with critical Besov regularity (2018)
  • Author(s): M. Fujita, N. Tomita
  • Journal Title: Proceedings of the American Mathematical Society Volume: 146 Issue: 2 Pages: 555-569
  • DOI: 10.1090/proc/13680
  • Peer Reviewed
• [Journal Article] Multilinear Fourier multipliers with minimal Sobolev regularity, II (2017)
  • Author(s): L. Grafakos, A. Miyachi, H.V. Nguyen, N. Tomita,
  • Journal Title: Journal of the Mathematical Society of Japan Volume: 69 Issue: 2 Pages: 529-562
  • DOI: 10.2969/jmsj/06920529
  • NAID: 130006887132
  • ISSN: 0025-5645, 1881-1167, 1881-2333
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Bilinear pseudo-differential operators with symbols in BS^m_{1,1} on Triebel-Lizorkin spaces (2017)
  • Author(s): K. Koezuka and N. Tomita
  • Journal Title: Journal of Fourier Analysis and Applications Volume: 未定 Issue: 1 Pages: 309-319
  • DOI: 10.1007/s00041-016-9518-2
  • Peer Reviewed
• [Presentation] S_{0,0} 型の双線形擬微分作用素について (2019)
  • Author(s): 冨田 直人
  • Organizer: 信州大学偏微分方程式研究集会
  • Invited
• [Presentation] Bilinear pseudo-differential operators with exotic symbols (2019)
  • Author(s): N. Tomita
  • Organizer: 12th International ISAAC Congress
  • Int'l Joint Research / Invited
• [Presentation] On the boundedness of bilinear pseudo-differential operators of S_{0,0}-type (2019)
  • Author(s): N. Tomita
  • Organizer: 2019 Taiwan Mathematical Society Annual Meeting
  • Int'l Joint Research / Invited
• [Presentation] On the $L^2 \times L^2 \to L^r$ boundedness of bilinear pseudo-differential operators of $S_{0,0}$-type (2019)
  • Author(s): N. Tomita
  • Organizer: NCTS Workshop on Harmonic Analysis (Taiwan)
  • Int'l Joint Research / Invited
• [Presentation] 双線形擬微分作用素とその応用 (2018)
  • Author(s): 冨田直人
  • Organizer: 非線形波動及び分散型方程式の研究
  • Invited
• [Presentation] Bilinear pseudo-differential operators with exotic symbols (2018)
  • Author(s): N. Tomita
  • Organizer: International Conference on Harmonic Analysis and its Applications (China)
  • Int'l Joint Research / Invited
• [Presentation] Bilinear pseudodifferential operators of type 1,1 and their application to the Kato-Ponce inequality (2017)
  • Author(s): N. Tomita
  • Organizer: 5nd East Asian Conference in Harmonic Analysis and Applications (China)
  • Int'l Joint Research / Invited
• [Presentation] Bilinear pseudodifferential operators of type 1,1 and their application to the Kato-Ponce inequality (2017)
  • Author(s): N. Tomita
  • Organizer: Harmonic Analysis and its Applications in Tokyo 2017
  • Int'l Joint Research / Invited
• [Presentation] Weighted norm inequalities for multilinear Fourier multipliers with critical Besov regularity (2016)
  • Author(s): N. Tomita
  • Organizer: International Workshop on Harmonic Analysis and its Applications in Beijing 2016
  • Place of Presentation: Beihang University (China)
  • Year and Date: 2016-09-29
  • Int'l Joint Research / Invited
• [Presentation] On the boundedness of multilinear Fourier multiplier operators (2016)
  • Author(s): 冨田 直人
  • Organizer: 調和解析と非線形偏微分方程式
  • Place of Presentation: 京都大学
  • Year and Date: 2016-07-04
  • Int'l Joint Research / Invited