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Study of global solutions to Fuchsian equations and local solutions to linear PDE

Research Project

Project/Area Number 15540156
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

OKADA Yasunori  Chiba University, Faculty of Sciences, Associate Professor, 理学部, 助教授 (60224028)

Co-Investigator(Kenkyū-buntansha) HINO Yoshiyuki  Chiba University, Faculty of Sciences, Professor, 理学部, 教授 (70004405)
ISHIMURA Ryuichi  Chiba University, Faculty of Sciences, Professor, 理学部, 教授 (10127970)
NAGISA Masaru  Chiba University, Faculty of Sciences, Professor, 理学部, 教授 (50189172)
TSUTSUI Toru  Chiba University, Faculty of Sciences, Assistant Professor, 理学部, 講師 (00197732)
Project Period (FY) 2003 – 2005
Project Status Completed (Fiscal Year 2005)
Budget Amount *help
¥2,900,000 (Direct Cost: ¥2,900,000)
Fiscal Year 2005: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2004: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2003: ¥1,000,000 (Direct Cost: ¥1,000,000)
KeywordsFuchsian equations / linear PDE / microlocal analysis / second microlocalization
Research Abstract

1.We tried second microlocal analysis for some class of Turrittin equations with lower terms of homogeneous type. In this case, similarly to the case of Mizohata equations, it became clear that fundamental solutions can be constructed using the theory of second microlocal Fourier transformations. Here we used the result about the growth order for the global solutions to some ordinary differential equations in the space where Fourier transforms live. In more special cases, the inverse Fourier transform satisfies some Fuchsian equations. We got the information of the Fuchsian equations, singularities, exponents, and some behaviors of their solutions.
2.On the other hand, we can consider the notion of microfunctions with holomorphic parameters on a product space of a real domain and a complex domain. They are represented as boundary values of holomorphic functions, and their boundary values define the notion of second microfunctions, and so on. The equations in 1 have microlocal ellipticity in rather small region, but we can construct their second microlocal solutions by the actions of their inverses to microfunctions with holomorphic parameters. The relation between this result and the theory of boundary values of pseudo-differential operators due to Kataoka will be a further subject.
3.A fundamental solution to linear PDE can be regarded as a kernel function of an integral transformation. We studied the kernels under an expectation that an integral transformation with kernel should be characterized by some notion similar to continuity, and got the kernel theorems in analytic category, by introducing notion of semi-continuity. Here we used the construction of complex kernels with curvilinear Radon transformations.

Report

(4 results)
  • 2005 Annual Research Report   Final Research Report Summary
  • 2004 Annual Research Report
  • 2003 Annual Research Report
  • Research Products

    (21 results)

All 2005 2004 Other

All Journal Article (19 results) Publications (2 results)

  • [Journal Article] Stability properties of Linear Volterra integrodiffenrential equations in a Banach space2005

    • Author(s)
      Y.Hino, S.Murakami
    • Journal Title

      Funkcialaj 48(3)

      Pages: 367-392

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Second Microlocalization, Regular Sequences, and Fourier Inverse Transforms2005

    • Author(s)
      O.Liess, Y.Okada, N.Tose
    • Journal Title

      数理解析研究所講究録 1412

      Pages: 13-21

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Stability properties of linear Volterra integrodifferential equations in a Banach space2005

    • Author(s)
      Y.Hino, S.Murakami
    • Journal Title

      Funkcialaj Eqvacioj 48(3)

      Pages: 367-392

    • NAID

      130000141321

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Second Microlocalization, Regular Sequences, and Fourier Inverse Transforms2005

    • Author(s)
      O.Liess, Y.Okada, N.Tose
    • Journal Title

      RIMS.Kokyuroku 1412

      Pages: 13-21

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Stability properties of linear Volterra integrodifferential equations in a Banach space2005

    • Author(s)
      Y.Hino, S.Murakami
    • Journal Title

      Funkcialaj Eqvacioj 48-3

      Pages: 367-392

    • NAID

      130000141321

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Second Microlocalization, Regular Sequences, and Fourier Inverse2005

    • Author(s)
      O.Liess, Y.Okada, N.Tose
    • Journal Title

      数理解析研究所講究録 1412

      Pages: 13-21

    • Related Report
      2004 Annual Research Report
  • [Journal Article] Numerical radius norm for bounded module maps and Schur multipliers2004

    • Author(s)
      T.Itoh, M.Nagisa
    • Journal Title

      Acta Sci. Math. (Szeged) 70(1-2)

      Pages: 195-211

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Some subsemigroups of extensions of C*-algebras2004

    • Author(s)
      Y.Katabami, M.Nagisa
    • Journal Title

      J. Math Soc. Japan 56(2)

      Pages: 561-569

    • NAID

      10013123140

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On the kernel theorem in hyperfunctions2004

    • Author(s)
      O.Liess, Y.Okada
    • Journal Title

      数理解析研究所講究録 1397

      Pages: 159-171

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary 2004 Annual Research Report
  • [Journal Article] Single generation and rank of C*-algebras2004

    • Author(s)
      M.Nagisa
    • Journal Title

      Operator algebras and applications, Adv. Stud. Pure Math., Math. Soc. Japan, Tokyo 38

      Pages: 135-143

    • NAID

      120007086759

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Triangle Fuchsian differential equations with apparent singularities2004

    • Author(s)
      H.Shiga, T.Tsutsui, J.Wolfart
    • Journal Title

      Osaka J. Math. 41

      Pages: 625-658

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Numerical radius norm for bounded module maps and Schur multipliers2004

    • Author(s)
      T.Itoh, M.Nagisa
    • Journal Title

      Acta Sci.Math.(Szeged) 70(1-2)

      Pages: 195-211

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Some subsemigroups of extensions of C*-algebras2004

    • Author(s)
      Y.Katabami, M.Nagisa
    • Journal Title

      J.Math.Soc.Japan 56(2)

      Pages: 561-569

    • NAID

      10013123140

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] On the kernel theorem in hyperfunctions2004

    • Author(s)
      O.Liess, Y.Okada
    • Journal Title

      RIMS.Kokyuroku 1397

      Pages: 159-171

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Single generation and rank of C*-algebras2004

    • Author(s)
      M.Nagisa
    • Journal Title

      Operator algebras and applications, Adv.Stud.Pure Math., Math.Soc.Japan, Tokyo 38

      Pages: 135-143

    • NAID

      120007086759

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Triangle Fuchsian differential equations with apparent singularities2004

    • Author(s)
      H.Shiga, T.Tsutsui, J.Wolfart
    • Journal Title

      Osaka J.Math. 41

      Pages: 625-658

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Transformations de Fourier-Sato et ope 'rateurs pseudo-diffe' rentiels non-locaux

    • Author(s)
      R.Ishimura
    • Journal Title

      Kyushu J. Math. (発表予定)

      Pages: 13-13

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Transformations de Fourier-Sato et ope' rateurs pseudo-diffe' rentiels non-locaux

    • Author(s)
      R.Ishimura
    • Journal Title

      Kyushu J.Math. (To appear)

      Pages: 13-13

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2005 Final Research Report Summary
  • [Journal Article] Transformations de Fourier-Sato et ope'rateurs pseudo-diffe'rentiels non-locaux

    • Author(s)
      R.Ishimura
    • Journal Title

      Kyushu J.Math. (発表予定)

      Pages: 13-13

    • Related Report
      2005 Annual Research Report
  • [Publications] Yoshiyuki Hino, Satoru Murakami: "Almost automorphic solutions for abstract functional differential equations"J.Math.Anal.Appl.. 286. 741-752 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] M.Nagisa: "Single generation and rank of C^*-algebras""Operator Algebras and Applications", Adv.Study in Pure Math., Math.Soc.Japan. 38. 135-143 (2004)

    • Related Report
      2003 Annual Research Report

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Published: 2003-04-01   Modified: 2016-04-21  

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