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Analysis of properties of the solutions to Scrodinger equations via canonical transforms

Research Project

Project/Area Number 18540176
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionOsaka University

Principal Investigator

SUGIMOTO Mitsuru  Osaka University, Dept. Math, Asso. Prof (60196756)

Co-Investigator(Kenkyū-buntansha) HOSHIRO Toshihiko  Univ. Hyogo, Dept. Math, Prof (40211544)
DOI Shin-ichi  Osaka Univ, Dept. Math, Prof (00243006)
HAYASHI Nakao  Osaka Univ, Dept. Math, Prof (30173016)
Project Period (FY) 2006 – 2007
Project Status Completed (Fiscal Year 2007)
Budget Amount *help
¥3,910,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥510,000)
Fiscal Year 2007: ¥2,210,000 (Direct Cost: ¥1,700,000、Indirect Cost: ¥510,000)
Fiscal Year 2006: ¥1,700,000 (Direct Cost: ¥1,700,000)
KeywordsFourier Integral operator / Scrodinger equation / smoothing effect / dispersive equation / non-linear problem / time-space estimate / modulation space / time-frequency analysis / 時間 / 自空間評価
Research Abstract

This research is an attempt to apply the method of canonical transform to various problems of Scrodinger or more general equations.
In the first year, we first established the theory of Fourier integral operators which is a main tool to realize our idea. To put it concretely, we gave the symbolic calculus for the composition of Fourier integral operators and pseudo- differential operators with their phase and amplitude functions in wider classes. By using this result, we got be able to discuss the mapping property of them in various kinds of function spaces like Sobolev or Bosov spaces.
In the second year, we developed some method to show the time-space estimates for dispersive equations. We found the fact that the comparison of some quantity of symbols implies the comparison of time-space estimates. We named it "comparison principle", and by using it, we developed some method which deduce the estimates in one or two space dimension to some simple model "estimate". Furthermore, if we combine the method of canonical transforms and the comparison principle, we can deduce time-space estimates for almost every dispersive equation to just a trivial estimate. We also investigated fundamental properties of modulation spaces, and apply them to prove some mapping properties of pseudo-differential equations.

Report

(3 results)
  • 2007 Annual Research Report   Final Research Report Summary
  • 2006 Annual Research Report
  • Research Products

    (16 results)

All 2008 2007 2006

All Journal Article (12 results) (of which Peer Reviewed: 4 results) Presentation (2 results) Book (2 results)

  • [Journal Article] A counterexample for boundedness of pseudo-differential operators on modulation spaces2008

    • Author(s)
      杉本 充(冨田直人との共著)
    • Journal Title

      Proc. Amer. Math. Soc. 136

      Pages: 1681-1690

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] A counterexample for boundedness of pseudo-differential operators on modulation spaces2008

    • Author(s)
      Mitsuru, Sugimoto, Naohito, Tomita
    • Journal Title

      Proc. Amer. Math. Soc 136

      Pages: 1681-1690

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] The dilation property of modulation spaces and their inclusion relation with Besov spaces2007

    • Author(s)
      杉本 充(冨田直人との共著)
    • Journal Title

      J. Funct. Anal. 248

      Pages: 79-106

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] The dilation property of modulation spaces and their inclusion relation with Besov spaces2007

    • Author(s)
      Mitsuru, Sugimoto, Naohito, Tomita
    • Journal Title

      J. Funct. Anal 248

      Pages: 79-106

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] A new aspect of the Lp-extension problem for inhomogeneous differential equations2007

    • Author(s)
      杉本 充
    • Journal Title

      Oper. Theory Adv. Appl. 172

      Pages: 153-159

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A smoothing property of Schrodinger equations in the critical case2006

    • Author(s)
      杉本 充(Ruzhanskyとの共著)
    • Journal Title

      Math. Ann. 335

      Pages: 645-673

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] A smoothing property of Schrudinger equations in the critical case2006

    • Author(s)
      Michael, Ruzhansky, Mitsuru, Sugimoto
    • Journal Title

      Math. Ann 335

      Pages: 645-673

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Global L2-boundedness theorems for a class of Fourier integral operators, Comm. Partial Differential Equations2006

    • Author(s)
      杉本 充 (Ruzhansky との共著)
    • Journal Title

      Comm. Partial Differential Equations 31巻

      Pages: 547-569

    • Related Report
      2006 Annual Research Report
  • [Journal Article] A smoothing property of Schrodinger equations in the critical case2006

    • Author(s)
      杉本 充 (Ruzhansky との共著)
    • Journal Title

      Math. Ann. 335巻

      Pages: 645-673

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Global calculus of Fourier integral operators, weighted estimates, and applications to global analysis of hyperbolic equations2006

    • Author(s)
      杉本 充 (Ruzhansky との共著)
    • Journal Title

      Oper. Theory Adv. Appl. 164巻

      Pages: 65-78

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Global boundedness theorems for Fourier integral operators associated with canonical transformations2006

    • Author(s)
      杉本 充 (Ruzhansky との共著)
    • Journal Title

      Proceedings of the International Conference on Harmonic Analysis and its Applications, (Yokohama Publishers) (臨時号)

      Pages: 65-75

    • Related Report
      2006 Annual Research Report
  • [Journal Article] A new aspect of the L^p-extension problem for inhomogeneous differential eauations.2006

    • Author(s)
      杉本 充
    • Journal Title

      Oper. Theory Adv. Appl. 172巻

      Pages: 153-159

    • Related Report
      2006 Annual Research Report
  • [Presentation] Smoothing estimates for dispersive equations via canonical transforms and comparison principle2007

    • Author(s)
      杉本 充
    • Organizer
      COE Seminar on Microlocal Analysis
    • Place of Presentation
      慶應義塾大学日吉キャンパス
    • Year and Date
      2007-08-03
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
  • [Presentation] Smoothing estimates for dispersive equations via canonical transforms and mparison principle2007

    • Author(s)
      Mitsuru, Sugimoto
    • Organizer
      COE Seminar on Microlocal Analysis
    • Place of Presentation
      Keio University
    • Year and Date
      2007-08-03
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Book] プリンストン解析学講義1「フーリエ解析入門」エリアス・M.スタイン, ラミ・シャカルチ著2007

    • Author(s)
      新井仁之, 杉本充, 高木啓行, 千原浩之, による共訳
    • Total Pages
      312
    • Publisher
      日本評論社
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Book] プリンストン解析学講義1「フーリエ解析入門」エリアス・M.スタイン, ラミ・シャカルチ著2007

    • Author(s)
      新井仁之, 杉本充, 高木啓行, 千原浩之 による共訳
    • Total Pages
      312
    • Publisher
      日本評論社
    • Related Report
      2007 Annual Research Report

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Published: 2006-04-01   Modified: 2016-04-21  

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