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2004 Fiscal Year Final Research Report Summary

Harmonic analysis in a domain with fractal boundary

Research Project

Project/Area Number 14540157
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

WATANABE Hisako  Ochanomizu University, Faculty of Science, Professor, 理学部, 教授 (70017193)

Co-Investigator(Kenkyū-buntansha) TAKEO Fukiko  Ochanomizu University, Faculty of Science, Professor, 理学部, 教授 (40109228)
YOSHIDA Hidenobu  Chiba Univ., Graduate School of Natural Sciences, Professor, 大学院・自然科学研究科, 教授 (60009280)
Project Period (FY) 2002 – 2004
Keywordsfractal lateral boundary / double layer heat potentials / Whitney decomposition / Besov spaces / Besov norms / maximal functions / uniform domains / boundedness of operators
Research Abstract

We consider the initial-boundary-value problems in a domain Ω=D×[O,T] with fractal lateral boundary S. It often occurs that an operator K on the Besov space B on the lateral boundary is bounded with respect to the Besov norms on S. We can prove the boundedness of an operator from B to B in the following method.
(1)We extend a function f defined on S to R×[O,T] by using an extension operator E.
(2)The Besov norm of f is estimated by the integral of |▽f(x)|×δ(x)^n over Ω, where δ(x) is the distance from x to S and n is a suitable number.
(3)Instead of the boundedness of K we prove the boundedness of an operator from a function space on Ω to a function space on the outside of Ω by using the maximal functions between Ω and the outside of Ω.
We proved the boundedness of an operator K, which is important to solve the Dirichlet problem by using double layer heat potentials.

  • Research Products

    (11 results)

All 2003 2002

All Journal Article (11 results)

  • [Journal Article] Estimates of the Besov norms on the fractal boundary and applications.2003

    • Author(s)
      H.Watanabe
    • Journal Title

      J.Math.Soc. Japan 55,3

      Pages: 565-589

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Beurling-Dahlberg-Sjogren type theorems for minimally thin sets in a cone2003

    • Author(s)
      H.Yoshida, I.Miyamoto, M.Yanagishita
    • Journal Title

      Canad.Math.Bull. 46

      Pages: 254-264

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Supercyclic and chaotic and translation semigroups2003

    • Author(s)
      F.Takeo, M.Matsui, M.Yamada
    • Journal Title

      Proc.Amer.Math.Soc. 131

      Pages: 3535-3546

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Estimates of the Besov norms on the fractal boundary and applications2003

    • Author(s)
      H.Watanabe
    • Journal Title

      J.Math.Soc.Japan 55-3

      Pages: 565-589

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Supercyclic and chaotic translation semigroups2003

    • Author(s)
      F.Takeo, M.Matsui, M.Yamada
    • Journal Title

      Proc.Amer.Math.Soc. 131

      Pages: 3535-3546

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Parabolic extension of lateral functions in a cylindrical domain2002

    • Author(s)
      H.Watanabe
    • Journal Title

      Natur.Sci.Rep.Ochanomizu Univ 53,2

      Pages: 21-34

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Two criterions of Wiener type for minimally thin sets and rarefied sets in a cone.2002

    • Author(s)
      H.Yoshida, I.Miyamoto
    • Journal Title

      J.Math.Soc. Japan 54

      Pages: 489-512

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] On chaotic semigroups generated by certain types of first-order differential operators2002

    • Author(s)
      F.Takeo, M.Matsui
    • Journal Title

      数理解析研究所講究録 1253

      Pages: 108-120

    • Description
      「研究成果報告書概要(和文)」より
  • [Journal Article] Parabolic extension of lateral functions in a cylindrical domain2002

    • Author(s)
      H.Watanabe
    • Journal Title

      Natur.Sci.Rep.Ochanomizu Univ. 53-2

      Pages: 21-34

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Two criterions of Wiener type for minimally thin sets and rarefied sets in a cone2002

    • Author(s)
      H.Yoshida, I.Miyamoto
    • Journal Title

      J.Math.Soc.Japan 54

      Pages: 488-512

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] On chaotic semigroups generated by certain types of first-order differential operators2002

    • Author(s)
      F.Takeo, M.Matsui
    • Journal Title

      RIMS 1253

      Pages: 108-120

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

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Published: 2006-07-11  

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