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Potential theoretic study for elliptic partial differential equations

Research Project

Project/Area Number 15K04929
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Basic analysis
Research InstitutionHiroshima University

Principal Investigator

SHIMOMURA TETSU  広島大学, 教育学研究科, 教授 (50294476)

Project Period (FY) 2015-04-01 – 2018-03-31
Project Status Completed (Fiscal Year 2017)
Budget Amount *help
¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000)
Fiscal Year 2017: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2016: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2015: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
Keywordsソボレフ関数 / 楕円型偏微分方程式
Outline of Final Research Achievements

Variable exponent Lebesgue spaces and Sobolev spaces were introduced to discuss nonlinear partial differential equations with non-standard growth condition. These spaces have attracted more and more attention in connection with the study of elasticity and electrorheological fluids. In this research, we studied the boundedness of the Hardy-Littlewood maximal operator on Musielak-Orlicz-Morrey spaces and grand Musielak-Orlicz-Morrey spaces. As an application of the boundedness of the maximal operator, we establish a generalization of Sobolev's inequality, Trudinger's exponential inequality and continuity for Riesz potentials of functions in Musielak-Orlicz-Morrey spaces and grand Musielak-Orlicz-Morrey spaces.

Report

(4 results)
  • 2017 Annual Research Report   Final Research Report ( PDF )
  • 2016 Research-status Report
  • 2015 Research-status Report
  • Research Products

    (26 results)

All 2017 2016 2015

All Journal Article (22 results) (of which Int'l Joint Research: 14 results,  Peer Reviewed: 22 results) Presentation (4 results) (of which Int'l Joint Research: 1 results,  Invited: 1 results)

  • [Journal Article] BOUNDEDNESS OF THE MAXIMAL OPERATOR ON MUSIELAK-ORLICZ-MORREY SPACES2017

    • Author(s)
      F. - Y. Maeda, T. Ohn, T. Shimomura
    • Journal Title

      Tohoku Mathematical Journal, Second Series

      Volume: 69 Issue: 4 Pages: 483-495

    • DOI

      10.2748/tmj/1512183626

    • ISSN
      0040-8735, 2186-585X
    • Year and Date
      2017-12-30
    • Related Report
      2017 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Trudinger's inequality and continuity for Riesz potential of functions in Orlicz spaces of two variable exponents over non-doubling measure spaces2017

    • Author(s)
      Sachihiro Kanemori, Takao Ohno and Tetsu Shimomura
    • Journal Title

      Kyoto J. Math.

      Volume: 57 Issue: 1 Pages: 79-96

    • DOI

      10.1215/21562261-3759522

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Boundary limits of monotone Sobolev functions in Musielak-Orlicz spaces on uniform domains in a metric space2017

    • Author(s)
      Takao Ohno and Tetsu Shimomura
    • Journal Title

      Kyoto J. Math.

      Volume: 57 Issue: 1 Pages: 147-164

    • DOI

      10.1215/21562261-3759549

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Boundedness of the generalized fractional integral operators on generalized Morrey spaces over metric measure spaces2017

    • Author(s)
      Y. Sawano, T. Shimomura
    • Journal Title

      Zeit. Anal. Anwend.

      Volume: 36 Issue: 2 Pages: 159-190

    • DOI

      10.4171/zaa/1584

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Generalized fractional integral operators over non-doubling metric measure spaces2017

    • Author(s)
      Y. Sawano, T. Shimomura
    • Journal Title

      Integral Transforms and Special Functions

      Volume: 28 Issue: 7 Pages: 534-546

    • DOI

      10.1080/10652469.2017.1318281

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Growth properties near the origin for generalized Riesz potentials2017

    • Author(s)
      Yoshihiro Mizuta, Takao Ohno, Tetsu Shimomura and Yusuke Yamauchi
    • Journal Title

      J. Math. Anal. Appl.

      Volume: 54 Issue: 1 Pages: 285-302

    • DOI

      10.1016/j.jmaa.2017.04.068

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Growth properties for Riesz potentials of functions in weighted variable L^p(・) spaces2017

    • Author(s)
      Y. Mizuta, T. Ohno, T. Shimomura
    • Journal Title

      Nonlinear Analysis

      Volume: 162 Pages: 49-75

    • DOI

      10.1016/j.na.2017.06.010

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Duality of non-homogeneous central Herz-Morrey-Musielak-Orlicz spaces2017

    • Author(s)
      F. - Y. Maeda, Y. Mizuta, T. Ohno, T. Shimomura
    • Journal Title

      Potential Anal.

      Volume: 47 Issue: 4 Pages: 447-460

    • DOI

      10.1007/s11118-017-9621-2

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Sobolev inequalities for Riesz potentials of functions in $L^{p(\cdot)}$ over non-doubling measure spaces2016

    • Author(s)
      T. Ohno, T. Shimomura
    • Journal Title

      Bull. Aust. Math. Soc.

      Volume: 93 Issue: 1 Pages: 128-136

    • DOI

      10.1017/s0004972715001331

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Musielak-Orlicz-Sobolev spaces with zero boundary values on metric measure spaces2016

    • Author(s)
      T. Ohno, T. Shimomura
    • Journal Title

      Czechoslovak Math. J.

      Volume: 66 Issue: 2 Pages: 371-394

    • DOI

      10.1007/s10587-016-0262-1

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Boundedness of generalized fractional integral operators from the Morrey space $L_{1,\phi}(X;\mu)$ to the Campanato space ${\mathcal L}_{1,\psi}(X;\mu)$ over non-doubling measure spaces2016

    • Author(s)
      Denny Ivanal Hakim, Y. Sawano, T. Shimomura
    • Journal Title

      Azerbaijan J. Math.

      Volume: 6 Pages: 117-127

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Some norm inequalities in Musielak-Orlicz spaces2016

    • Author(s)
      F. - Y. Maeda, Y. Sawano, T. Shimomura
    • Journal Title

      Ann. Acad. Sci. Fenn. Math.

      Volume: 41 Pages: 721-744

    • DOI

      10.5186/aasfm.2016.4148

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Trudinger's inequality and continuity for Riesz potentials of functions in grand Musielak-Orlicz-Morrey spaces over non-doubling metric measure spaces2016

    • Author(s)
      T. Ohno, T. Shimomura
    • Journal Title

      Kyoto J. Math.

      Volume: 56 Issue: 3 Pages: 633-653

    • DOI

      10.1215/21562261-3600211

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Boundedness of maximal operators and Sobolev's inequality on non-homogeneous central Musielak-Orlicz-Morrey spaces2016

    • Author(s)
      T. Ohno, T. Shimomura
    • Journal Title

      Mediterr. J. Math.

      Volume: 13 Pages: 3341-3357

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Growth properties of Musielak-Orlicz integral means for Riesz potentials2015

    • Author(s)
      F. - Y. Maeda, Y. Mizuta, T. Shimomura
    • Journal Title

      Nonlinear Anal.

      Volume: 112 Pages: 69-83

    • DOI

      10.1016/j.na.2014.09.012

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Boundary limits of monotone Sobolev functions with variable exponent on uniform domains in a metric space2015

    • Author(s)
      T. Futamura, T. Ohno, T. Shimomura
    • Journal Title

      Rev. Mat. Complut.

      Volume: 28 Issue: 1 Pages: 31-48

    • DOI

      10.1007/s13163-014-0154-6

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Sobolev type and Trudinger type inequality on grand Musielak-Orlicz-Morrey spaces2015

    • Author(s)
      F. - Y. Maeda, Y. Mizuta, T. Ohno, T. Shimomura
    • Journal Title

      Ann. Acad. Sci. Fenn. Math.

      Volume: 40 Pages: 403-426

    • DOI

      10.5186/aasfm.2015.4027

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Weighted Morrey spaces of variable exponent and Riesz potentials2015

    • Author(s)
      Y. Mizuta, T. Shimomura
    • Journal Title

      Math. Nachr.

      Volume: 288 Issue: 8-9 Pages: 984-1002

    • DOI

      10.1002/mana.201400032

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Optimal estimates for the fractional Hardy operator2015

    • Author(s)
      Y. Mizuta, Ales Nekvinda, T. Shimomura
    • Journal Title

      Studia Math.

      Volume: 227 Issue: 1 Pages: 1-19

    • DOI

      10.4064/sm227-1-1

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Musielak-Orlicz-Sobolev spaces on metric measure spaces2015

    • Author(s)
      T. Ohno, T. Shimomura
    • Journal Title

      Czechoslovak Math. J.

      Volume: 140 Issue: 2 Pages: 435-474

    • DOI

      10.1007/s10587-015-0187-0

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Boundedness of maximal operators and Sobolev's theorem for non-homogeneous central Morrey spaces of variable exponent2015

    • Author(s)
      Y. Mizuta, T. Ohno, T. Shimomura
    • Journal Title

      Hokkaido Math. J.

      Volume: 44 Pages: 185-201

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Exponential integrability for logarithmic potentials of functions in generalized Lebesgue spaces $L(\log L)^{q(\cdot)}$ over non-doubling measure spaces2015

    • Author(s)
      S. Kanemori, T. Ohno, T. Shimomura
    • Journal Title

      Taiwanese J. Math.

      Volume: 19 Issue: 6 Pages: 1795-1803

    • DOI

      10.11650/tjm.19.2015.5564

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] Musielak-Orlicz空間におけるSobolevの不等式について2017

    • Author(s)
      下村 哲
    • Organizer
      実解析学シンポジウム 2017
    • Related Report
      2017 Annual Research Report
  • [Presentation] Boundary behavior of monotone Sobolev functions on John domains2017

    • Author(s)
      下村 哲
    • Organizer
      第60回関数論シンポジウム
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] Sobolev inequalities for Musielak-Orlicz spaces2017

    • Author(s)
      Tetsu Shimomura
    • Organizer
      Nonstandard growth phenomena 2017
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Optimal estimates for the fractional Hardy operator2016

    • Author(s)
      下村哲
    • Organizer
      日本数学会秋季総合分科会
    • Place of Presentation
      関西大学
    • Related Report
      2016 Research-status Report

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Published: 2015-04-16   Modified: 2019-03-29  

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