Sharp conditions on the boundedness of multilinear operators

• Project/Area Number: 24740088
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Basic analysis
• Research Institution: Osaka University
• Principal Investigator: TOMITA Naohito 大阪大学, 理学(系)研究科(研究院), 准教授 (10437337)
• Project Period (FY): 2012-04-01 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000); Fiscal Year 2015: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2014: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2013: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2012: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: 多重線形作用素 / フーリエマルチプライヤー / 擬微分作用素 / flag paraproduct

Outline of Final Research Achievements

I studied the boundedness properties of bilinear Fourier multiplier and pseudo-differential operators. To assure the boundedness, the smoothness condition for Fourier multipliers in the sense of Sobolev spaces and the order of pseudo-differential operators were determined. The sharpness of these results was also studied.

Research Products

• [Journal Article] Estimates for trilinear flag paraproducts on L^{infty} and Hardy spaces (2015)
  • Author(s): Akihiko Miyachi and Naohito Tomita
  • Journal Title: Mathematische Zeitschrift Volume: 282 Issue: 1-2 Pages: 577-613
  • DOI: 10.1007/s00209-015-1554-0
  • Peer Reviewed / Open Access
• [Journal Article] Boundedness criterion for bilinear Fourier multiplier operators (2014)
  • Author(s): A. Miyachi and N. Tomita
  • Journal Title: Tohoku Mathematical Journal Volume: 66 Issue: 1 Pages: 55-76
  • DOI: 10.2748/tmj/1396875662
  • NAID: 130007953667
  • Peer Reviewed
• [Journal Article] A counterexample to weighted estimates for multilinear Fourier multipliers with Sobolev regularity (2014)
  • Author(s): M. Fujita and N. Tomita
  • Journal Title: Journal of Mathematical Analysis and Applications Volume: 409 Issue: 2 Pages: 630-636
  • DOI: 10.1016/j.jmaa.2013.07.041
  • Peer Reviewed
• [Journal Article] Minimal smoothness conditions for bilinear Fourier multipliers (2013)
  • Author(s): A. Miyachi and N. Tomita
  • Journal Title: Revista Matematica Iberoamericana Volume: 29 Issue: 2 Pages: 495-530
  • DOI: 10.4171/rmi/728
  • Peer Reviewed
• [Journal Article] Calderon-Vaillancourt type theorem for bilinear operators (2013)
  • Author(s): A. Miyachi and N. Tomita
  • Journal Title: Indiana University Mathematics Journal Volume: 62 Issue: 4 Pages: 1165-1201
  • DOI: 10.1512/iumj.2013.62.5059
  • Peer Reviewed
• [Journal Article] On multilinear Fourier multipliers of limited smoothness (2013)
  • Author(s): L. Grafakos, A. Miyachi, N. Tomita
  • Journal Title: Canadian Journal of Mathematics Volume: 65 Issue: 2 Pages: 299-330
  • DOI: 10.4153/cjm-2012-025-9
  • Peer Reviewed
• [Journal Article] Weighted norm inequalities for multilinear Fourier multipliers (2012)
  • Author(s): M. Fujita and N. Tomita
  • Journal Title: Transactions of the American Mathematical Society Volume: 364 Issue: 12 Pages: 6335-6353
  • DOI: 10.1090/s0002-9947-2012-05700-x
  • Peer Reviewed
• [Journal Article] A remark on multilinear Fourier multipliers satisfying Besov estimates (2012)
  • Author(s): N. Tomita
  • Journal Title: RIMS Kokyuroku Bessatsu Volume: 33 Pages: 111-121
  • Peer Reviewed
• [Presentation] On the boundedness of multilinear Fourier multiplier operators (2016)
  • Author(s): N. Tomita
  • Organizer: Harmonic analysis and its applications in Matsumoto 2016, Winter
  • Place of Presentation: 信州大学
  • Year and Date: 2016-02-15
  • Invited
• [Presentation] 多重線形フーリエマルチプライヤー作用素の有界性について (2015)
  • Author(s): 冨田 直人
  • Organizer: 浜松偏微分方程式研究集会
  • Place of Presentation: 静岡大学
  • Year and Date: 2015-12-22
  • Invited
• [Presentation] On the boundedness of multilinear Fourier multiplier operators (2015)
  • Author(s): N. Tomita
  • Organizer: Harmonic Analysis and its Applications in Tokyo 2015
  • Place of Presentation: 早稲田大学
  • Year and Date: 2015-11-27
  • Int'l Joint Research / Invited
• [Presentation] Flag paraproducts on Hardy spaces (2014)
  • Author(s): N. Tomita
  • Organizer: 2nd East Asian Conference in Harmonic Analysis and Applications
  • Place of Presentation: Mudanjiang Normal University (Mudanjiang, China)
  • Year and Date: 2014-07-11 – 2014-07-14
• [Presentation] 特異性の強い多重線形フーリエマルチプライヤーについて (2014)
  • Author(s): 冨田直人
  • Organizer: 南大阪応用数学セミナー
  • Place of Presentation: 大阪市立大学
  • Year and Date: 2014-05-31
  • Invited
• [Presentation] 双線形擬微分作用素に対するCalderon-Vaillancourt 型の定理について (2013)
  • Author(s): 冨田直人
  • Organizer: 線形および非線形分散型方程式の研究
  • Place of Presentation: 京都大学
  • Invited
• [Presentation] 双線形フーリエマルチプライヤー作用素の有界性について (2013)
  • Author(s): 冨田直人
  • Organizer: 日本数学会 秋季総合分科会，特別講演(函数方程式分科会)
  • Place of Presentation: 愛媛大学
  • Invited
• [Presentation] Smoothness conditions for bilinear Fourier multipliers (2013)
  • Author(s): N. Tomita
  • Organizer: 1st East Asian Conference in Harmonic Analysis and Applications
  • Place of Presentation: Seoul National University
• [Presentation] Sharp estimates for bilinear Fourier multipliers (2012)
  • Author(s): N. Tomita
  • Organizer: Fourier Analysis and Pseudo-Differential Operators
  • Place of Presentation: Aalto University (Finland)
• [Presentation] Sharp estimates for bilinear Fourier multipliers (2012)
  • Author(s): 冨田直人
  • Organizer: 実函数論函数解析学合同シンポジウム
  • Place of Presentation: 東京理科大学
  • Invited
• [Presentation] Calderon-Vaillancourt type theorem for bilinear operators (2012)
  • Author(s): N. Tomita
  • Organizer: Harmonic Analysis and its Applications at Tokyo 2012
  • Place of Presentation: 首都大学東京
  • Invited
• [Presentation] 双線形擬微分作用素に対するCalderon-Vaillancourt型の定理について (2012)
  • Author(s): 冨田直人
  • Organizer: 調和解析セミナー
  • Place of Presentation: 東京大学