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Study on null forms in global space-time in the framework of equalities

Research Project

Project/Area Number 16K13771
Research Category

Grant-in-Aid for Challenging Exploratory Research

Allocation TypeMulti-year Fund
Research Field Mathematical analysis
Research InstitutionWaseda University

Principal Investigator

Ozawa Tohru  早稲田大学, 理工学術院, 教授 (70204196)

Co-Investigator(Kenkyū-buntansha) 田中 和永  早稲田大学, 理工学術院, 教授 (20188288)
BEZ NEAL  埼玉大学, 理工学研究科, 准教授 (30729843)
Project Period (FY) 2016-04-01 – 2020-03-31
Project Status Completed (Fiscal Year 2019)
Budget Amount *help
¥3,640,000 (Direct Cost: ¥2,800,000、Indirect Cost: ¥840,000)
Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2017: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2016: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Keywords関数方程式 / 調和解析 / 実解析 / 変分解析
Outline of Final Research Achievements

Stability of trace theorems on the sphere is studied as the most fundamental subject in the research of null forms in global space-time. We have established the desired optimal inequalities for the stability theory and given its characterization from the viewpoint of duality. Regarding the Hardy and Rellich inequalities, we have formulated their equality framework with explicit remainder terms, therely we were able to recast the associated best constants and extremizers in a direct and explicit understanding. This provides a new method, independent of implicit arguments of contradiction and compactness.

Academic Significance and Societal Importance of the Research Achievements

零形式の時空大域的研究の基盤を揺ぎ無いものとするとともに、函数不等式の従来の理解を、不等式ではなく、等式の枠組みで具体的に与えることができた。

Report

(5 results)
  • 2019 Annual Research Report   Final Research Report ( PDF )
  • 2018 Research-status Report
  • 2017 Research-status Report
  • 2016 Research-status Report
  • Research Products

    (63 results)

All 2020 2019 2018 2017 2016 Other

All Int'l Joint Research (3 results) Journal Article (19 results) (of which Int'l Joint Research: 15 results,  Peer Reviewed: 19 results,  Open Access: 4 results,  Acknowledgement Compliant: 5 results) Presentation (32 results) (of which Int'l Joint Research: 29 results,  Invited: 31 results) Book (3 results) Remarks (3 results) Funded Workshop (3 results)

  • [Int'l Joint Research] バーミンガム大学(英国)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] バーミンガム大学(英国)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] バーミンガム大学(英国)

    • Related Report
      2016 Research-status Report
  • [Journal Article] Hardy type inequalities with spherical derivatives2020

    • Author(s)
      Bez Neal、 Machihara Shuji、 Ozawa Tohru
    • Journal Title

      SN Partial Differ. Equ. Appl.

      Volume: 1, Issue 1 Issue: 1

    • DOI

      10.1007/s42985-019-0001-1

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Semi-classical states for the nonlinear Choquard equations: existence, multiplicity and concentration at a potential well2019

    • Author(s)
      Cingolani Silvia、Tanaka Kazunaga
    • Journal Title

      Rev. Mat. Iberoam.

      Volume: 35 Issue: 6 Pages: 1885-1924

    • DOI

      10.4171/rmi/1105

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] A note on deformation argument for $L^2$ normalized solutions of nonlinear Schr\"odinger equations and systems2019

    • Author(s)
      Ikoma Norihisa, Tanaka Kazunaga
    • Journal Title

      Adv. Differential Equations

      Volume: 24 Pages: 609-646

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Nonlinear Scalar Field Equations with $L^2$ Constraint: Mountain Pass and Symmetric Mountain Pass Approaches2019

    • Author(s)
      Hirata Jun, Tanaka Kazunaga
    • Journal Title

      Advanced Nonlinear Studies

      Volume: 19 Issue: 2 Pages: 263-290

    • DOI

      10.1515/ans-2018-2039

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] Stability of Trace Theorems on the Sphere2018

    • Author(s)
      Neal Bez, Chris Jeavons, Tohru Ozawa, Mitsuru Sugimoto
    • Journal Title

      The Journal of Geometric Analysis

      Volume: 28 Issue: 2 Pages: 1456-1476

    • DOI

      10.1007/s12220-017-9870-8

    • NAID

      120006496423

    • Related Report
      2018 Research-status Report 2017 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] On the lifespan of strong solutions to the periodic derivative nonlinear Schr\"odinger equation2018

    • Author(s)
      Fujiwara Kazumasa, Ozawa Tohru
    • Journal Title

      Evolution Equations & Control Theory

      Volume: 7 Issue: 2 Pages: 275-280

    • DOI

      10.3934/eect.2018013

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Lifespan of periodic solutions to nonlinear Schr\"odinger equations2018

    • Author(s)
      Fujiwara Kazumasa, Ozawa Tohru
    • Journal Title

      "Nonlinear Wave and Dispersive Equations," RIMS Kokyuroku 2093(2018)

      Volume: 2093 Pages: 38-46

    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Remarks on the Hardy type inequalities with remainder terms in the framework of equalities2018

    • Author(s)
      Machihara Shuji, Ozawa Tohru, Wadade Hidemitsu
    • Journal Title

      Adv. Studies Pure Math.

      Volume: 印刷中

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] Smoothing Estimates for the Kinetic Transport Equation at the Critical Regularity2018

    • Author(s)
      Bez Neal, Cunanan Jayson, Lee Sanghyuk
    • Journal Title

      SIAM Journal on Mathematical Analysis

      Volume: 50 Issue: 2 Pages: 2280-2294

    • DOI

      10.1137/17m1148852

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Remarks on the Rellich inequality2017

    • Author(s)
      S. Machihara, T. Ozawa, H. Wadade
    • Journal Title

      Mathematische Zeitschrift

      Volume: 印刷中 Issue: 3-4 Pages: 1367-1373

    • DOI

      10.1007/s00209-016-1805-8

    • NAID

      120005982988

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] Multi-bump solutions for logarithmic Schr"odinger equations2017

    • Author(s)
      K. Tanaka, C. Zhang
    • Journal Title

      Calculus of Variations and Partial Differential Equations

      Volume: 56 Issue: 2

    • DOI

      10.1007/s00526-017-1122-z

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Multiple complex-valued solutions for nonlinear magnetic Schr"odinger equations2017

    • Author(s)
      S. Cingolani, L. Jeanjean, K. Tanaka
    • Journal Title

      Journal of Fixed Point Theory and Applications

      Volume: 19 Issue: 1 Pages: 37-66

    • DOI

      10.1007/s11784-016-0347-3

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Remarks on the Clark theorem2017

    • Author(s)
      Guosheng Jiang, Tanaka Kazunaga, Zhang Chengxiang
    • Journal Title

      J. Nonlinear Convex Anal.

      Volume: 18 Pages: 1421-1434

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] Ground states for semi-relativistic Schr¨odinger-Poisson-Slater energy2017

    • Author(s)
      J. Bellazzini, T. Ozawa, N. Visciglia
    • Journal Title

      Funkcialaj Ekvacioj

      Volume: 印刷中

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Higher order fractional Leibniz rule2017

    • Author(s)
      K. Fujiwara, V. Georgiev, T. Ozawa
    • Journal Title

      J. Fourier Anal.Appl.

      Volume: 印刷中

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Behaviour of the Brascamp-Lieb constant2017

    • Author(s)
      Jonathan Bennett, Neal Bez, Michael Cowling, Taryn Flock
    • Journal Title

      Bulletin of the London Mathematical Society

      Volume: - Issue: 3 Pages: 512-518

    • DOI

      10.1112/blms.12049

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Stability of the Brascamp-Lieb constant and applications2017

    • Author(s)
      J. Bennett, N. Bez, T. Flock, S. Lee
    • Journal Title

      The American Journal of Mathematics

      Volume: 印刷中

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
  • [Journal Article] Estimates for the kinetic transport equation in hyperbolic Sobolev spaces2017

    • Author(s)
      J. Bennett, N. Bez, S. Gutierrez, S. Lee
    • Journal Title

      Journal de Math'ematiques Pures et Appliqu'ees

      Volume: 印刷中

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Sharp lower bounds for Coulomb energy2016

    • Author(s)
      J. Bellazzini, M. Ghimenti, T. Ozawa
    • Journal Title

      Math. Research Letters

      Volume: 23 Issue: 3 Pages: 621-632

    • DOI

      10.4310/mrl.2016.v23.n3.a2

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] Spherical Harmonics and Hardy’s equality2020

    • Author(s)
      Machihara Shuji
    • Organizer
      Recent Progress on Dispersive and Wave Equations
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] 微分型相互作用をもつ非線型シュレディンガー方程式の自己相似解2019

    • Author(s)
      Ozawa Tohru
    • Organizer
      第9回岐阜数理科学研究会
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] A new deformation argument for $L^2$ normalized solutions of nonlinear Schr\"odiger equations and systems2019

    • Author(s)
      Tanaka Kazunaga
    • Organizer
      非線性微分方程学木検討会(Workshop on nonlinear differential equations
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 微分型シュレディンガー方程式の自己相似解2019

    • Author(s)
      Ozawa Tohru
    • Organizer
      PDE Workshop in Miyazaki
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] A new deformation theory under the (PSP) condition2019

    • Author(s)
      Tanaka Kazunaga
    • Organizer
      第13回 非線形偏微分方程式と変分問題
    • Related Report
      2018 Research-status Report
  • [Presentation] Lifespan estimates of solutions to NLS without gauge invariance2018

    • Author(s)
      Ozawa Tohru
    • Organizer
      PDE Seminar
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Lifespan of blowup solutions of DNLS type equation on the torus2018

    • Author(s)
      Ozawa Tohru
    • Organizer
      PDE Seminar
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] A new minimax approach to nonlinear scalar field equations with L 2 constraint2018

    • Author(s)
      Tanaka Kazunaga
    • Organizer
      International Conference on Variational Methods (ICVAM-4)
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Smoothing estimates for the kinetic transport equation at the critical regularity2018

    • Author(s)
      Bez Neal
    • Organizer
      保存則をもつ偏微分方程式の解の正則性、特異性および漸近挙動の研究
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On the kinetic transport equation2018

    • Author(s)
      Bez Neal
    • Organizer
      調和解析と非線形偏微分方程式
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Strichartz estimates for orthonormal systems of initial data2018

    • Author(s)
      Bez Neal
    • Organizer
      International Workshop on Fundamental Problems in Mathematical and Theoretical Physics
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Mixed-norm estimates for multipliers associated with some hypersurfaces2018

    • Author(s)
      Bez Neal
    • Organizer
      Harmonic Analysis and PDE Workshop
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Geometric estimates arising in the analysis of Zakharov systems2018

    • Author(s)
      Bez Neal
    • Organizer
      2018日本数学会秋季総合分科会
    • Related Report
      2018 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Lifespan of periodic solutions to derivative nonlinear Schr\"odinger equations2018

    • Author(s)
      Ozawa Tohru
    • Organizer
      Nonlinear Dispersive Equations in Kumamoto, 2018
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Lifespan of blowup solutions to the nonlinear Schr\"odinger equations on torus2018

    • Author(s)
      Ozawa Tohru
    • Organizer
      Hyperbolic Partial Differential Equations and Related Topics-in honor of the 60th birthday of Professor Tokio Matsuyama-
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] On improved Hardy inequalities2018

    • Author(s)
      Ozawa Tohru
    • Organizer
      Workshop on Harmonic analysis and Nonlinear Evolution Equations
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Minimax methods for scalar field equations with L^2-constraint2018

    • Author(s)
      Tanaka Kazunaga
    • Organizer
      第12回 非線形偏微分方程式と変分問題
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] A new minimax approach to nonliear scalar field equations with L^2 constraint2018

    • Author(s)
      Tanaka Kazunaga
    • Organizer
      Seminari di Matematica
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Stability of the Brascamp-Lieb inequality2017

    • Author(s)
      Neal Bez
    • Organizer
      Analysis Seminar
    • Place of Presentation
      University of Edinburgh (イギリス・エジンバラ)
    • Year and Date
      2017-03-20
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Smoothing estimates for the kinetic transport equation via the cone multiplier2017

    • Author(s)
      Neal Bez
    • Organizer
      Harmonic Analysis Workshop
    • Place of Presentation
      金沢大学 (石川県・金沢市)
    • Year and Date
      2017-03-05
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Smoothing estimates for velocity averages via the cone multiplier2017

    • Author(s)
      Neal Bez
    • Organizer
      Harmonic Analysis and Applications Conference
    • Place of Presentation
      Seoul National University (韓国・ソウル)
    • Year and Date
      2017-03-01
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Lifespan of periodic solutions to nonlinear Schr\"odinger equations2017

    • Author(s)
      Ozawa Tohru
    • Organizer
      RIMS Workshop Nonlinear Wave and Dispersive Equations
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Blowup solutions for the derivative nonlinear Schr\"odinger equation on torus2017

    • Author(s)
      Tohru Ozawa
    • Organizer
      Recent topics on PDEs
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Multi-bump solutions for logarithmic Schr\"odinger equations2017

    • Author(s)
      Tanaka Kazunaga
    • Organizer
      信州大学偏微分方程式研究集会
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Singular perturbation problem for generalized Choquard equations2017

    • Author(s)
      Tanaka Kazunaga
    • Organizer
      the Third Congress of the PRIMA, the Pacific Rim Mathematical Association
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Strichartz estimates for orthonormal systems of initial data2017

    • Author(s)
      Bez Neal
    • Organizer
      Nonlinear Partial Differential Equations for Future
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Stability of the Brascamp-Lieb constant and applications2016

    • Author(s)
      Neal Bez
    • Organizer
      International Conference for the 70th Anniversary of the Korean Mathematical Society
    • Place of Presentation
      Seoul National University (韓国・ソウル)
    • Year and Date
      2016-10-22
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Uncertainty relations in the framework of equalities2016

    • Author(s)
      Tohru Ozawa
    • Organizer
      International Conference for the 70th Anniversary of Korean Mathematical Society
    • Place of Presentation
      Seoul National University (韓国・ソウル)
    • Year and Date
      2016-10-20
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Estimates for the kinetic transport equation in hyperbolic Sobolev spaces2016

    • Author(s)
      Neal Bez
    • Organizer
      Analysis Seminar
    • Place of Presentation
      Seoul National University (韓国・ソウル)
    • Year and Date
      2016-10-19
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Uncertainty relations in the framework of equalities2016

    • Author(s)
      Tohru Ozawa
    • Organizer
      New trends in Partial Differential Equations
    • Place of Presentation
      Centro di Ricerca Matematica Ennio De Giorgi (イタリア・ピサ)
    • Year and Date
      2016-10-03
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Higher order fractional Leibniz estimates2016

    • Author(s)
      Tohru Ozawa
    • Organizer
      Nonlinear Wave and Dispersive Equations, Kyoto 2016
    • Place of Presentation
      京都大学 (京都府・京都市左京区)
    • Year and Date
      2016-09-06
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 等式の枠組から観た不確定性関係2016

    • Author(s)
      Tohru Ozawa
    • Organizer
      微分方程式セミナー Seminar of Differential Equations
    • Place of Presentation
      大阪大学 数学教室 大セミナー室(E301) (大阪府・豊中市)
    • Year and Date
      2016-06-10
    • Related Report
      2016 Research-status Report
    • Int'l Joint Research / Invited
  • [Book] New Tools for Nonlinear PDEs and Application, Trends in Mathematics2019

    • Author(s)
      M. D'Abbicco, M. R. Ebert, V. Georgiev, and T. Ozawa (Eds.)
    • Total Pages
      390
    • Publisher
      Birkh\"auser
    • ISBN
      9783030109363
    • Related Report
      2019 Annual Research Report
  • [Book] Asymptotic analysis of nonlinear dispersive and wave equations2019

    • Author(s)
      K. Kato, T. Ogawa, and T. Ozawa (Eds.)
    • Total Pages
      419
    • Publisher
      Mathematical Society of Japan
    • ISBN
      9784864970815
    • Related Report
      2019 Annual Research Report
  • [Book] 数理物理学としての微分方程式序論2016

    • Author(s)
      小澤徹
    • Total Pages
      192
    • Publisher
      サイエンス社
    • Related Report
      2016 Research-status Report
  • [Remarks] 早稲田大学 理工学術院 先進理工学部 応用物理学科 小澤 徹研究室

    • URL

      http://www.ozawa.phys.waseda.ac.jp/index2.html

    • Related Report
      2019 Annual Research Report
  • [Remarks] 小澤 徹研究室

    • URL

      http://www.ozawa.phys.waseda.ac.jp/index2.html

    • Related Report
      2018 Research-status Report
  • [Remarks] 小澤徹研究室

    • URL

      http://www.ozawa.phys.waseda.ac.jp/index2.html

    • Related Report
      2017 Research-status Report 2016 Research-status Report
  • [Funded Workshop] Waseda Workshop on Partial Differential Equations 20192019

    • Related Report
      2018 Research-status Report
  • [Funded Workshop] 非線型科学コロキウム2018

    • Related Report
      2018 Research-status Report
  • [Funded Workshop] "Applied Analysis"2017

    • Related Report
      2017 Research-status Report

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Published: 2016-04-21   Modified: 2022-02-22  

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