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Synthetic Study for Real Analysis and its Applications

Research Project

Project/Area Number 10440048
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionOKAYAMA UNIVERSITY

Principal Investigator

SATO Ryotaro  Faculty of Science, Okayama University, Professor, 理学部, 教授 (50077913)

Co-Investigator(Kenkyū-buntansha) KITA Hiro-o  Oita University, Faculty of Education, Professor, 教育福祉科学部, 教授 (20224941)
TANAKA Naoki  Faculty of Science, Okayama University, Associate Professor, 理学部, 助教授 (00207119)
HASEGAWA Shigeru  Shibaura Institute of Technology, Faculty of Technology, Professor, 工学部, 教授 (50052832)
TAKAHASHI Yasuji  Okayama Prefectural University, Faculty of Computer Science and System Engineering, Professor, 情報工学部, 教授 (30001853)
Project Period (FY) 1998 – 1999
Project Status Completed (Fiscal Year 1999)
Budget Amount *help
¥6,600,000 (Direct Cost: ¥6,600,000)
Fiscal Year 1999: ¥3,600,000 (Direct Cost: ¥3,600,000)
Fiscal Year 1998: ¥3,000,000 (Direct Cost: ¥3,000,000)
KeywordsErgodic theorems / positive operators / maximal ergodic functions / measure preserving transformations / semigroups of operators / almost everywhere convergence
Research Abstract

We considered pointwise ergodic thorems for n-dimensional semigroups of operators on vector-valued functions spaces. It is difficult to prove pointwise ergodic theorems for these semigroups. The main reason is that the estimate of the maximal ergodic functions is not obtained (except for the one-dimensional semigroup case). This remains still an open problem. But, in case the semigroup consists of positive operators, the estimate can be obtained by using the reduction method of Dunford and Schwartz. This method is applied to complex-valued function spaces, and Emilion has succeeded to prove a general local ergodic theorem by this method. Unfortunately, this is not applied to vector-valued function spaces, because the linear modulus of an operator does not exists in general for an operator on vector-valued function spaces. About this problem, we extended the notion of linear modulus of an operator, and then proved, under some suitable norm conditions on operator, that the pointwise ergodic theorem holds for n-dimensional semigroups of operators. Also we proved that this method remains valid to prove the local ergodic theorem for n-dimensional semigroups of operators on vector-valued functions spaces.

Report

(3 results)
  • 1999 Annual Research Report   Final Research Report Summary
  • 1998 Annual Research Report
  • Research Products

    (22 results)

All Other

All Publications (22 results)

  • [Publications] Ryotaro Sato: "On a vector-valued local erogodic theorem"Studia Math.. 132. 285-298 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Ryotaro Sato: "Vector-valued ergodic theorems"Colloq. Math.. 79. 193-202 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Ryotaro Sato: "A general differentiation theorem"Colloq. Math.. (発表予定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Naoki Tanaka: "Global solutions of abstract quasi-linear"Israel J. Math.. 110. 219-252 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Yasuji Takahashi et al.: "An extension of Hlawka's inequality"Math. Inequialities Appl.. 3. 63-67 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Ryotaro Sato: "Vector-valued differentiation theorems for multi-parameter additive processes in Lp-spaces"Positivity. 2. 1-18 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Ryotaro Sato: "On a vector-valued local ergodic theorem in L"Studia Math.. 132. 285-2998 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Ryotaro Sato: "Vector-valued ergodic theorems for multiparameter additive processes"Colloq. Math.. 79. 193-202 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Ryotaro Sato: "A general differentiation theorem for superadditive processes"Colloq. Math.. (in press).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Ryotaro Sato: "On the weighted ergodic properties of Lamperti operators"Math. J. Okayama Univ.. (in press).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Naoki Tanaka: "Generation of linear evolution operators"Proc. Amer. Math. Soc.. (in press).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Sin-Ei Takahashi, Yasuji Takahashi and Shuhei Wada: "An extension of Hlawka's inequality"Math. Inequalities and Appl.. 3. 63-67 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Hiro-o Kita: "Integrability properties of the maximal operators on partial sums of Fourier series in Orlicz spaces"Math. Nachr.. 193. 57-74 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      1999 Final Research Report Summary
  • [Publications] Ryotaro Sato: "On a vector-valued local ergodic theorem"Studia Math.. 132. 285-298 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Ryotaro Sato: "Vector-valued ergodic theorems"Colloq.Math.. 79. 193-202 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Ryotaro Sato: "A general differentiation theorem"Colloq.Math.. (発表予定).

    • Related Report
      1999 Annual Research Report
  • [Publications] Naoki Tanaka: "Global solutions of abstract quasi-linear"Israel J.Math.. 110. 219-252 (1999)

    • Related Report
      1999 Annual Research Report
  • [Publications] Yasuji Takahashi et al.: "An extension of Hlawka's inequality"Math.Inequalities Appl.. 3. 63-67 (2000)

    • Related Report
      1999 Annual Research Report
  • [Publications] R.Sato: "Vector valued differentiation theorems" Positibity. 2. 1-18 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] N.Tanaka: "Global solutions of abstract evolution equations" Israel J.Math.(発表予定).

    • Related Report
      1998 Annual Research Report
  • [Publications] H.Kita: "Integrability properties of the maximal operators" Math.Nachr.193. 57-74 (1998)

    • Related Report
      1998 Annual Research Report
  • [Publications] Y.Takahashi: "Von Neumann-Jordan constant and Banach spaces" Nihonkai J.Math.9. 155-169 (1998)

    • Related Report
      1998 Annual Research Report

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Published: 1998-04-01   Modified: 2016-04-21  

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