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2005 Fiscal Year Final Research Report Summary

Study on prehomogeneous vector spaces and micro-local analysis

Research Project

Project/Area Number 15340042
Research Category

Grant-in-Aid for Scientific Research (B)

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

Principal Investigator

MURO Masakazu  Gifu University, Faculty of Engineering, Professor, 工学部, 教授 (70127934)

Co-Investigator(Kenkyū-buntansha) SHIGA Kiyoshi  Gifu University, Faculty of Engineering, Professor, 工学部, 教授 (10022683)
KOBAYASHI Takako  Gifu University, Faculty of Engineering, Associate Professor, 工学部, 助教授 (40252126)
SEKIGUCHI Jiro  Tokyo Noukou University, Graduate School, Professor, 大学院・共生科学技術研究部, 教授 (30117717)
OSHIMA Toshio  Tokyo Noukou University, Graduate School, Professor, 大学院・数理科学研究科, 教授 (50011721)
GYOJA Akihiko  Tokyo Noukou University, Graduate School, Professor, 大学院・多元数理科学研究科, 教授 (50116026)
Project Period (FY) 2003 – 2005
Keywordsprehomogeneous vector space / micro-local analysis / Representation of Lie groups / Invariant Theory / Invariant hyperfunctions / Invariant differential equations / Group Theory / fundamental solution
Research Abstract

In the period of the research, we mainly have been studying invariant differential equations on prehomogeneous vector spaces. The invariant differential equations on prehomogeneous vector spaces we are studying are linear and of constant coefficients, so they are the easiest differential operators to analyze. However, the concrete analysis for the specific differential operators does not seem to be so easy. Indeed, except for the wave operators, the concrete analysis for the specific hyperbolic differential operators with higher order is less well understood. We approached the analysis of invariant differential operators on prehomogeneous vector spaces through the problem of the determination of the support and the singularity spectra of the fundamental solution. In particular, the invariant differential operators on the prehomogeneous vector spaces of commutative parabolic type are hyperbolic differential operators. It is the most orthodox way for the calculation of the singularity propagation to determine the exact singularity spectra of the fundamental solution. We have succeeded to define the singularity propagation sets of the differential operators and determine them.
The most remarkable result is that we have established the "Huygens principle" for the singularity propagation and discover the example for which the "Huygens principle" holds by the explicit computation. We verified that the "Huygens principle" for the singularity propagation holds for the invariant differential operators on prehomogeneous vector spaces. It is an interesting and challenging problem whether the "Huygens principle" is valid for other differential operators.

  • Research Products

    (11 results)

All 2005 2004 2003

All Journal Article (11 results)

  • [Journal Article] 不変超関数とその応用---超局所解析による計算2005

    • Author(s)
      室 政和
    • Journal Title

      第44回実関数論・関数解析学合同シンポジウム会議録 44

      Pages: 124-146

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] A quantization of conjugacy classes of matrices2005

    • Author(s)
      Toshio Oshima
    • Journal Title

      Adv. Math. 196

      Pages: 124-146

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Construction of invariant hyperfunctions and its application2005

    • Author(s)
      M Muro
    • Journal Title

      Proceeding of the 44th Joint Conference of Real Function Theory and Functional Analysis

      Pages: 92-107

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] A quantization of conjugacy classes of matrices2005

    • Author(s)
      Toshio Oshima
    • Journal Title

      Adv.Math. Vol.196

      Pages: 124-146

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Algorithmic construction of hyperfunction solutions to invariant differential equations on the space of real symmetric matrices2004

    • Author(s)
      M.Muro
    • Journal Title

      J. Lie Theory 14

      Pages: 111-140

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Regular polyhedral groups and reflection groups of rank four2004

    • Author(s)
      Mitsuo Kato, Jiro Sekiguchi
    • Journal Title

      European Journal of Combinatorics 25

      Pages: 565-577

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Algorithmic construction of hyperfunction solutions to invariant differential equations on the space of real symmetric matrices2004

    • Author(s)
      M Muro
    • Journal Title

      J.Lie Theory Vol.14

      Pages: 111-140

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Regular polyhedral groups and reflection groups of rank four2004

    • Author(s)
      Mitsuo Kato, Jiro Sekiguchi
    • Journal Title

      European Journal of Combinatorics Vol.25

      Pages: 565-577

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Fundamental solutions, Cauchy problems and Huygens principle for invariant differential operators on prehomogeneous vector spaces of commutative parabolic type2003

    • Author(s)
      M.Muro
    • Journal Title

      Surikaisekikenkyusho Kokyuroku 1348

      Pages: 61-74

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Fatou's theorems and Hardy-type spaces for eigenfunctions of the invariant differential operators on symmetric spaces2003

    • Author(s)
      Ben Said, Salem, Oshima, T., Shimeno, N.
    • Journal Title

      Int. Math. Res. Not. 16

      Pages: 915-931

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Fundamental solutions, Cauchy problems and Huygens principle for invariant differential operators on prehomogeneous vector spaces of commutative parabolic type2003

    • Author(s)
      M Mum
    • Journal Title

      Surikaisekikenkyusho Kokyuroku Vol.1348

      Pages: 61-74

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2007-12-13  

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