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Theory and applications of BMO and related function spaces on spaces of homogeneous type

Research Project

Project/Area Number 11640165
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

NAKAI Eiichi  Osaka Kyoiku Univ., Faculty of Education, Associate Professor, 教育学部, 助教授 (60259900)

Co-Investigator(Kenkyū-buntansha) TANAKA Hidenori  Osaka Kyoiku Univ., Faculty of Education, Associate Professor, 教育学部, 助教授 (60192176)
FUJII Masatoshi  Osaka Kyoiku Univ., Faculty of Education, Professor, 教育学部, 教授 (10030462)
CHODA Marie  Osaka Kyoiku Univ., Faculty of Education, Professor, 教育学部, 教授 (80030378)
SOBUKAWA Takuya  Okayama Univ., Faculty of Education, Department of Mathematics Education, Associate Professor, 教育学部, 助教授 (60252946)
IZUMISAWA Masataka  Tokai Univ., School of Science, Department of Mathematical Sciences, Professor, 理学部, 教授 (50108445)
Project Period (FY) 1999 – 2001
Project Status Completed (Fiscal Year 2001)
Budget Amount *help
¥2,500,000 (Direct Cost: ¥2,500,000)
Fiscal Year 2001: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2000: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 1999: ¥900,000 (Direct Cost: ¥900,000)
Keywordsbounded mean oscillation / space of homogeneous type / pointwise multiplier / fractional integral / Riesz potential / Orlicz space / Campanato space / Morrey space
Research Abstract

1. The theory of pointwise multipliers on the function space Lp is well known. In this research, we have developed this theory on several function spaces ; Lorentz, Orlicz, Morrey, BMO, and Campanato spaces. The pointwise multiplier is simple and basic. So we can examine properties of spaces of homogeneous type by studying it.
2. The boundedness of singular integral operators and the Riesz potential are useful for the theory of partial differential equations and are studied by many authors. In particular, the boundedness of the Riesz potential (fractional integral) from L^p to L^q is well known as the Hardy-Littlewood-Sobolev theorem. In this research, we introduce generalized fractional integrals and extend this boundedness on several function spaces ; Orlicz, BMO_φ, Campanato, Morrey, weak-Orlicz and generalized Hardy spaces.
3. We have developed the method to redefine the quasi-distance of the space of homogeneous type on which the function space is not change, and we have the following as applications :
(1) We have other results for the theory of pointwise multipliers on Campanato spaces.
(2) On several operators used in the theory of partial differential equations, for example Kohn Laplacian, some results on the normal space of homogeneous type are adaptable to general spaces of homogeneous type.
(3) On the fractional integral and derivative, some results on the normal space of homogeneous type are adaptable to general spaces of homogeneous type.

Report

(4 results)
  • 2001 Annual Research Report   Final Research Report Summary
  • 2000 Annual Research Report
  • 1999 Annual Research Report
  • Research Products

    (30 results)

All Other

All Publications (30 results)

  • [Publications] Eiichi Nakai: "On generalized fractional integrals"数理解析研究所講究録. 1137. 61-70 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Eiichi Nakai: "On generalized fractional integrals in the Orlicz spaces"Proceedings of the Second ISAAC Congress, Kluwer Academic Publishers B. V. Netherland-U. S. A.. 1. 75-81 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Eiichi Nakai: "A characterization of pointwise multipliers on the Moller spaces"Scientiae Mathematicae. 3. 445-454 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Eiichi Nakai, Hironori Sumitomo: "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
      2001 Final Research Report Summary
  • [Publications] Eiichi Nakai: "On generalized fractional integrals in the Orlicz 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
      2001 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
      2001 Final Research Report Summary
  • [Publications] Chikako Harada, Eiichi Nakai: "The square partial sums of the Fourier transform of radial functions in three dimensions"Scientiae Mathematicae Japonicae Online. 5. 329-339 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Eiichi Nakai: "On generalized fractional integrals on the weak Orlicz spaces, BMO_φ, the Morrey spaces and the Campanato spaces"Proceedings of the Conference on Function Spaces, Interpolation Theory, and related topics, Lund University, Sweden. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Eiichi Nakai: "Hardy spaces and preduals of Campanato spaces"京都大学数理解析研究所講究録. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Eiichi Nakai: "On generalized fractional integrals"Surikaisekikenkyusho Kokyuroku. No. 1137. 61-70 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Eiichi Nakai: "On generalized fractional integrals in the Orlicz spaces"Proceedings of the Second ISAAC Congress, Kluwer Academic Publishers B. V. Netherland - U.S.A.. 75-81 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Eiichi Nakai: "A characterization of pointwise multipliers on the Scientiae Mathematicae Morrey spaces"Vol.3. 445-454 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Eiichi Nakai and Hironori Sumitomo: "On generalized Riesz potentials and spaces of some smooth functions"Scientiae Mathematical Japonicae. Vol.54 (Scientiae Mathematicae Japonicae Online, Vol.4 (2001), 891-900). 463-472 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Eiichi Nakai: "On generalized fractional integrals in the Orlicz spaces on spaces of homogeneous type"Scientiae Mathematicae Japonicae. Vol.54 (Scientiae Mathematicae Japonicae Online, Vol.4 (2001), 901-915). 473-487 (2001)

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

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Chikako Harada and Eiichi Nakai: "The square partial sums of the Fourier transform of radial functions in three dimensions"Scientiae Mathematicae Japonicae Online. Vol.5 (to appear). 329-339 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Eiichi Nakai: "Un generalized fractional integrals on the weak Orlicz spaces, BMO_φ, the Morrey spaces and the Campanato spaces"Proceedings of the Conference on Function Spaces, Interpolation Theory, and related topics in honor of Jaak Peetre on his 65th birthday, Lund University Sweden. (to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2001 Final Research Report Summary
  • [Publications] Eiichi Nakai: "Hardy spaces and preduals of Campanato spaces"Surikaisekikenkyusho Kokyuroku. (to appear).

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

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

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

    • Related Report
      2001 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 Online. 5. 329-339 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] Eiichi Nakai: "On generalized fractional integrals of the weak Orlicz spaces, BMO_φ, the Morrey spaces and the Campanato spaces"Proceedings of the Conference on Function Spaces, Interpolation Theory, and related topics, Lund University, Sweden. (印刷中).

    • Related Report
      2001 Annual Research Report
  • [Publications] Eiichi Nakai: "Hardy spaces and preduals of Campanato spaces"京都大学数理解析研究所講究録. (印刷中).

    • Related Report
      2001 Annual Research Report
  • [Publications] Eiichi Nakai: "On generalized fractional integrals"数理解析研究所講究録. 1137. 61-70 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Eiichi Nakai: "On generalized fractional integrals in the Orlicz spaces"Proceedings of the Second ISAAC Congress,Kluwer Academic Publishers B.V.Netherland-U.S.A.. 1. 75-81 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Eiichi Nakai : "A characterization of pointwise multipliers on the Morrey spaces"Scientiae Mathematicae. 3. 445-454 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] Eiichi Nakai: "On Generalized fractional integrals"Taiwanese Journal of Mathematics. 9(印刷中). (2001)

    • Related Report
      2000 Annual Research Report
  • [Publications] Eiichi Nakai: "On generalized fractional integrals"京都大学 数理解析研究所 講究録. (印刷中). (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] Eiichi Nakai: "On generalized fractional integrals in the Orlicz spaces"Proceedings of the second Congress ISAAC. (印刷中). (2000)

    • Related Report
      1999 Annual Research Report

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Published: 1999-04-01   Modified: 2016-04-21  

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