• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to previous page

Weight theory related to harmonic analysis on groups -in harmony with representation theory

Research Project

Project/Area Number 13640190
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

KAWAZOE Takeshi  Keio University, Faculty of Policy Management, Professor, 総合政策学部, 教授 (90152959)

Co-Investigator(Kenkyū-buntansha) NAKAI Eiichi  Osaka Kyoiku University, Course for School Teacher, Associate Professor, 教育学部, 助教授 (60259900)
Project Period (FY) 2001 – 2003
Project Status Completed (Fiscal Year 2003)
Budget Amount *help
¥3,400,000 (Direct Cost: ¥3,400,000)
Fiscal Year 2003: ¥1,300,000 (Direct Cost: ¥1,300,000)
Fiscal Year 2002: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2001: ¥1,100,000 (Direct Cost: ¥1,100,000)
Keywordssemisimple Lie group / real Hardy space / Abel transform / fractional integral and derivative / Jaccobi transform / ハーディ空間 / 分数積分 / フーリエ・ヤコビ変換 / 逆変換公式 / 解析接続 / 最大関数 / 複素ヤコビ変換 / 非等質空間
Research Abstract

In this research we investigated the usage of fractional integrals in real analysis and also in harmonic analysis on semisimple Lie groups. E. Nakai, the investigator, studied boundedness of the fractional integral operators on various function spaces, especially, he generalized the operators and function spaces. T. Kawazoe, the head investigator, applied the theory of fractional integrals to construct real Hardy spaces on semisimple Lie groups. In this sense, we attained our aim of this research -in harmony with representation theory and real analysis.
The theory of real Hardy spaces was generalized on the so-called space of homogeneous type satisfying the doubling condition, however, little works were done on the space of non-homogeneous type. In this research we took real rank one semisimple Lie groups G as an example of the space of non-homogeneous type, and introduced real Hardy spaces for K-bi-invariant functions. This corresponds to harmonic analysis on Fourier-Jaccobi transforms … More and on the weight theory with exponential growth order. We first, by using the Abel transform, reduced the Fourier-Jaccobi analysis to Euclidean one on the real line R. Since the Abel transform was written by fractional integrals, the method of real analysis related to fractional integrals were useful in this analysis. Next, five extended the theory of fractional integrals and derivatives and then characterized real Hardy spaces defined by maximal functions and atoms. Our main theorem is the following. Let W be the modified Abel transform with Exp(ρx)-multiplication and V the inverse of W. Then, we see H1 (G//K) ⊂V (H1(R)), where H1(G//K) and HI(R) are respectively the real Hardy spaces on G and R defined by the radial maximal functions. Moreover, let A1(G//K)+ denote the atomic Hardy space on G defined by restricting the moment condition of atoms only for ones with redius <1. Then it follows that H1(G//K)= V(H1(R))∩A1(G//K)+. In the proof we applied some methods in real analysis related to fractional integrals and atomic decompositions. Less

Report

(4 results)
  • 2003 Annual Research Report   Final Research Report Summary
  • 2002 Annual Research Report
  • 2001 Annual Research Report
  • Research Products

    (33 results)

All Other

All Publications (33 results)

  • [Publications] Takeshi Kawazoe: "On the inversion formula of Jacobi transform"Keio Research Report. 7. 1-10 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Eiichi Nakai: "On generalized Riesz potentials and spaces of some smooth functions"Scientiae Mathematicae Japonicae. 54. 463-472 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Eiichi Nakai: "On generalized fractional integrals in the Orlicz spaces on spaces of homogeneous type"Scientiae Mathematicae Japonicae. 54. 473-487 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Eiichi Nakai: "On generalized fractional integrals"Taiwanese Journal of Mathematics. 5. 587-602 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Takeshi Kawazoe: "On a weak L1 property of maximal operators on non-compact semisimple Lie grpups"Tokyo J.Math.. 25. 165-179 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Chikako Harada: "The square partial sums of the Fourier transform of radial functions in three dimensions"Scientiae Mathematicae Japonicae. 55. 467-477 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Eiichi Nakai: "On generalized fractional integrals on the weak Orlicz spaces, BMOφ the Morrey spaces and the Campanato spaces, Function spaces"Interpolation theory and related topics (Walter de Gruyter, Berlin). 389-401 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Eiichi Nakai: "Hardy spaces and preduals of Campanato spaces"調和・解析関数空間と線形作用素II(京都大学数理解析研究所講究録). 1277. 66-77 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Takeshi Kawazoe: "Heat kernel and Hardy's theorem for Jacobi transform"Chin.Ann.Math.. 24. 359-366 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Eiichi Nakai: "Tangential boundary behavior of the Poisson integrals of functions in the potential space with the Orlicz norm"Scientiae Mathematicae Japonicae. (to appear). (2004)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Takeshi Kawazoe: "Fractional calculus and analytic continuation of the complex Fourier-Jacobi transform"Tokyo J.Math.. (to appear). (2004)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Takeshi Kawazoe: "On the inversion formula of Jacobi transform"Keio Research Report. 7. 1-10 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Eiichi Nakai: "On generalized Riesz potentials and spaces of some smooth functions"Scientiae Mathematicae Japonicae. 54. 463-472 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Eiichi Nakai: "On generalized fractional integrals in the Orlicz spaces on spaces of homogeneous type"Scientiae Mathematicae Japonicae. 54. 473-487 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Eiichi Nakai: "On generalized fractional integrals"Taiwanese Journal of Mathematics. 5. 587-602 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Takeshi Kawazoe: "On a weak L1 property of maximal operators on non-compact semisimple Lie grpups"Tokyo J.Math.. 25. 165-179 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Chikako Harada: "The square partial sums of the Fourier transform of radial functions in three dimensions"Scientiae Mathematicae Japonicae. 55. 467-477 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Eiichi Nakai: "On generalized fractional integrals on the weak Orlicz spaces, BMOφ, the Morrey spaces and the Campanato spaces, Function spaces"Interpolation theory and related topics (Walter de Gruyter, Berlin). 389-401 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Eiichi Nakai: "Hardy spaces and preduals of Campanato spaces"調和・解析関数空間と線形作用素II(京都大学数理解析研究所講究録). 1277. 66-77 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Takeshi Kawazoe: "Heat kernel and Hardy's theorem for Jacobi transform"Chin.Ann.Math.. 24. 359-366 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Eiichi Nakai: "Tangential boundary behavior of the Poisson integrals of functions in the potential space with the Orlicz norm"Scientiae Mathematicae Japonicae. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Takeshi Kawazoe: "Fractional calculus and analytic continuation of the complex Fourier-Jacobi transform"Tokyo J.Math.. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] Takeshi Kawazoe: "Heat kernel and Hardy's theorem for Jacobi transform"Chin.Ann.Math.. 24. 359-366 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] Eiichi Nakai: "Tangential boundary behavior of the Poisson integrals of functions in the potential space with the Orlicz norm"Scientiae Mathematicae Japonicae. (To appear). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] Takeshi Kawazoe: "Fractional calculus and analytic continuation of the complex Fourier-Jacobi transform"Tokyo J.Math.. (To appear). (2004)

    • Related Report
      2003 Annual Research Report
  • [Publications] Takeshi Kawazoe: "On a weak L1 property of maximal operators on non-compact semisimpie Lie groups"Tokyo J. Math.. 25(1). 165-179 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Chikako Harada, Eiichi Nakai: "The square partial sums of the Fourier transform of radial functions in three dimensions"Scientiae Mathematicae Japonicae. 55. 467-477 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] Eiichi Nakai: "On generalized fractional integrals on the weak Orliczspaces, BMO_φ, the Morrey spaces and the Campanato spaces"Proceedings of the International Conference on "Function Spaces, Interpolation Theory and related topics. 389-401 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Kawazoe: "On the inversion formula of Jacobi transform"Keio Research Report. 7. 1-10 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] E.Nakai: "On generalized Riesz potentials and spaces of some smooth functions"Scientiae Mathematicae Japonicae. 54. 463-472 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] E.Nakai: "On generalized fractional integrals in the Orlicz spaces"Scientiae Mathematicae Japonicae. 54. 473-487 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] E.Nakai: "On generalized fractional integrals"Taiwanese Journal of Mathematics. 5. 587-602 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] T.Kawazoe: "On a weak L^1property of maximal operators on non-comapct semisimple Lie groups"Tokyo J.Math.. (発表予定). (2002)

    • Related Report
      2001 Annual Research Report

URL: 

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

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi