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Global properties of differential operators of subdeterminantal type and integral geometry on symmetric spaces

Research Project

Project/Area Number 13640203
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionUniversity of Tsukuba

Principal Investigator

KAKEHI Tomoyuki  University of Tsukuba, Institute of Mathematics, Associate Professor, 数学系, 助教授 (70231248)

Co-Investigator(Kenkyū-buntansha) TAIRA Kazuaki  University of Tsukuba, Institute of Mathematics, Professor, 数学系, 教授 (90016163)
SASAKI Tateaki  University of Tsukuba, Institute of Mathematics, Professor, 数学系, 教授 (80087436)
KAJITANI Kunihiko  University of Tsukuba, Institute of Mathematics, Professor, 数学系, 教授 (00026262)
NAITO Satoshi  University of Tsukuba, Institute of Mathematics, Associate Professor, 数学系, 助教授 (60252160)
MIYAMOTO Masahiko  University of Tsukuba, Institute of Mathematics, Professor, 数学系, 教授 (30125356)
若林 誠一郎  筑波大学, 数学系, 教授 (10015894)
竹内 潔  筑波大学, 数学系, 講師 (70281160)
土居 伸一  筑波大学, 数学系, 助教授 (00243006)
Project Period (FY) 2001 – 2002
Project Status Completed (Fiscal Year 2002)
Budget Amount *help
¥4,000,000 (Direct Cost: ¥4,000,000)
Fiscal Year 2002: ¥1,500,000 (Direct Cost: ¥1,500,000)
Fiscal Year 2001: ¥2,500,000 (Direct Cost: ¥2,500,000)
KeywordsRadon transform / integral geometry / symmetric space / range characterization / inversion formula / パフィアン型微分作用素 / 平滑化作用 / アファイングラスマン多様体 / グラスマン多様体 / パフィアン / 不変微分作用素
Research Abstract

1. Pfaffian type operators and Radon transforms on affine Grassmann manifolds : Let G(d, n) be the affine Grassmann manifolds of all d-dimensional planes in R^n". Then the Radon transform R^p_q is defined as the transform from smooth functions on G(p,n) to smooth functions on G(q,n] arising from the inclusion incidence relation. Then our results are stated as follows. (1) In the case p < q. Let s and r be the rank of G(p, n) (resp. G(q, n) ). We assume that s < r. Then the range of R^p_q is characterized as the kernel of a single Pfaffian type invariant differenial operator of order 2s + 2. (2) In the case p < q. We assume that s 【less than or equal】 r. Then the inversion formula for R^p_q is given as DR^p_qR^p_q = I, where D is the reproducing operator consisting of Pfaffian type operators. (3) In the case p > q. We assume that s < r. Then the range of R^p_q is characterized as the kernel of an invariant system of differential equations of order s + 1, which consists of two different kinds of Pfaffians. This research was done in collaboration with F. Gonzalez.
2. Sobolev estimates for Radon transforms : Basically a Radon transform is an integration of a function over a submanifold. So it is expected that a Radon transform regularizes a function to some extent, and in fact, it was shown by Strichartz that the q-plane transform R^0_4 maps a function on L^2 to a funtion on H^<(9)/(2)> the Sobolev space of order 9/2. In this case, the gain of regularity is proportional to the demension of the fiber of the corresponding double fibration. However, in the case of R^p_q for general p and q, we discovered that R^p_q does not regularize a function so much in the sense that the gain of regularity is no longer proportional to the dimension of the fiber.

Report

(3 results)
  • 2002 Annual Research Report   Final Research Report Summary
  • 2001 Annual Research Report
  • Research Products

    (27 results)

All Other

All Publications (27 results)

  • [Publications] T.Kakehi (with F.Gonzalez): "Pfaffian systems and Radon transforms on affine Grassmann manifolds"Mathematische Annalen. (in press).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] S.Doi (with A.Iwatsuka, T.Mine): "The uniqueness of the integrated density of states for the Schrodinger operators with magnetic fields"Math. Z.. 237,no.2. 335-371 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K.Kajitani (with S.Wakabayashi, K.Yagdjian): "The hyperbolic operators with the characteristics vanishing with the different speeds"Osaka J. Mat.. 39,no.2. 447-485 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] S.Naito: "Twining character formula of Kac-Wakimoto type for affine Lie algebras"Represent. Theory. 6. 70-100 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T.Sasaki (with A.Terui): "Durand-Kerner method for the real roots"Japan J. Indust. Appl. Math.. 19,no.1. 19-38 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K.Taira: "Semilinear elliptic boundary-value problems in combustion theory"Proc. Roy. Soc. Edinburgh Sect. A. 132,no.6. 1453-1476 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K.Taira: "Introduction to diffusive logistic equations in population dynamics"Korean J. Comput. Appl. Math.. 9,no.2. 289-347 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K.Takeuchi: "Microlocal vanishing cycles and ramified Cauchy problems in the Nilsson class"Compositio Math.. 125,no.1. 111-127 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T. Kakehi, F. Gonzalez: "Pfaffian systems and Radon transforms on affine Grassmann manifolds"Mathematische Annalen. in press.

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] S. Doi (with A. Iwatsuka and T. Mine): "The uniqueness of the integrated density of states for the Schrodinger operators with magnetic fields"Math. Z.. 237 no. 2. 335-371 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K. Kajitani, S. Wakabayashi, K. Yagdjian: "The hyperbolic operators with the characteristics vanishing with the different speeds"Osaka J. Math.. 39 no. 2. 447-485 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] S. Naito: "Twining character formula of Kac-Wakimoto type for affine Lie algebras"Theory. 6. 70-100 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T. Sasaki, A. Terui.: "Durand-Kerner method for the real roots"Japan J. Indust. Appl. Math.. 19 no. 1. 19-38 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K. Taira: "Semilinear elliptic boundary-value problems in combustion theory"Proc. Roy. Soc. Edinburgh Sect. A. 132 no. 6. 1453-1476 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K. Taira: "Introduction to diffusive logistic equations in population dynamics"Korean J. Comput. Appl. Math.. 9 no. 2. 289-347 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K. Takeuchi: "Microlocal vanishing cycles and ramified Cauchy problems in the Nilsson class"Compositio Math.. 125 no. 1. 111-127 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] F.Gonzalez, T.Kakehi: "Pfaffian systems and Radon transforms on affine Grassmann manifolds"Mathematische Annalen. (発表予定).

    • Related Report
      2002 Annual Research Report
  • [Publications] Y.Dan, K.Kajitani: "Smoothing effect and exponential time decay of solutions of solutions of Schrodinger equations"Proc. Japan Acad. Ser. A Math. Sci. 78巻6号. 92-95 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Sasaki, A.Terui: "Durand Kerner method for the real roots"Japan Journal of Indust. Appl. Math.. 19巻1号. 19-38 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] K.Taira: "Diffusive logistic equations in population dynamics"Advances in Differential Equations. 7巻2号. 237-256 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Miyamoto: "3-state Potts model and automorphisms of vertex operator algebras of order 3"Journal of Algebra. 239巻1号. 56-76 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] S.Wakabayashi et al.: "The hyperbolic operators with the characteristics vanishing with the different speeds"Osaka Journal of Mathematics. 39巻2号. 447-485 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Taira, Kazuaki: "Diffuslve logistic equations in population dynamics"Achances in Differential Equations. 7巻 2号. 237-256 (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] Miyamoto, Masahiko: "3-state potts model and automorphism of vertex operator algchras of order 3"Journal of Algebra. 239巻 1号. 56-76 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Doi, Shin-ichi: "The unlqueness of the integrated density of states for the schrbdiuger opemtors with magnetic fields"Mathematische Zeitschrift. 237巻 2号. 335-371 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Naito, Satoshi: "Character formula Kac-Wakimoto type for generalized Kac-Moody algebras"Journal of Pure and Applied Algebra. 166巻 1-2号. 105-123 (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] Takeuchi, Kiyoshi: "Microlocal vanishing cycles and ramified Cauchy problems in the Nilsson class"Compositio Mathematica. 125巻 1号. 111-127 (2001)

    • Related Report
      2001 Annual Research Report

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Published: 2001-04-01   Modified: 2016-04-21  

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