Potential theoretic study for elliptic partial differential equations

• Project/Area Number: 15K04929
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Hiroshima 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: ¥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.

Research Products

• [Journal Article] BOUNDEDNESS OF THE MAXIMAL OPERATOR ON MUSIELAK-ORLICZ-MORREY SPACES (2017)
  • 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
  • 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 spaces (2017)
  • 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
  • Peer Reviewed
• [Journal Article] Boundary limits of monotone Sobolev functions in Musielak-Orlicz spaces on uniform domains in a metric space (2017)
  • Author(s): Takao Ohno and Tetsu Shimomura
  • Journal Title: Kyoto J. Math. Volume: 57 Issue: 1 Pages: 147-164
  • DOI: 10.1215/21562261-3759549
  • Peer Reviewed
• [Journal Article] Boundedness of the generalized fractional integral operators on generalized Morrey spaces over metric measure spaces (2017)
  • Author(s): Y. Sawano, T. Shimomura
  • Journal Title: Zeit. Anal. Anwend. Volume: 36 Issue: 2 Pages: 159-190
  • DOI: 10.4171/zaa/1584
  • Peer Reviewed
• [Journal Article] Generalized fractional integral operators over non-doubling metric measure spaces (2017)
  • 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
  • Peer Reviewed
• [Journal Article] Growth properties near the origin for generalized Riesz potentials (2017)
  • 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
  • Peer Reviewed
• [Journal Article] Growth properties for Riesz potentials of functions in weighted variable L^p(・) spaces (2017)
  • 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
  • Peer Reviewed
• [Journal Article] Duality of non-homogeneous central Herz-Morrey-Musielak-Orlicz spaces (2017)
  • 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
  • Peer Reviewed
• [Journal Article] Sobolev inequalities for Riesz potentials of functions in $L^{p(\cdot)}$ over non-doubling measure spaces (2016)
  • Author(s): T. Ohno, T. Shimomura
  • Journal Title: Bull. Aust. Math. Soc. Volume: 93 Issue: 1 Pages: 128-136
  • DOI: 10.1017/s0004972715001331
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Musielak-Orlicz-Sobolev spaces with zero boundary values on metric measure spaces (2016)
  • 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
  • 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 spaces (2016)
  • Author(s): Denny Ivanal Hakim, Y. Sawano, T. Shimomura
  • Journal Title: Azerbaijan J. Math. Volume: 6 Pages: 117-127
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Some norm inequalities in Musielak-Orlicz spaces (2016)
  • 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
  • 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 spaces (2016)
  • Author(s): T. Ohno, T. Shimomura
  • Journal Title: Kyoto J. Math. Volume: 56 Issue: 3 Pages: 633-653
  • DOI: 10.1215/21562261-3600211
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Boundedness of maximal operators and Sobolev's inequality on non-homogeneous central Musielak-Orlicz-Morrey spaces (2016)
  • Author(s): T. Ohno, T. Shimomura
  • Journal Title: Mediterr. J. Math. Volume: 13 Pages: 3341-3357
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Growth properties of Musielak-Orlicz integral means for Riesz potentials (2015)
  • 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
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Boundary limits of monotone Sobolev functions with variable exponent on uniform domains in a metric space (2015)
  • 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
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Sobolev type and Trudinger type inequality on grand Musielak-Orlicz-Morrey spaces (2015)
  • 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
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Weighted Morrey spaces of variable exponent and Riesz potentials (2015)
  • Author(s): Y. Mizuta, T. Shimomura
  • Journal Title: Math. Nachr. Volume: 288 Issue: 8-9 Pages: 984-1002
  • DOI: 10.1002/mana.201400032
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Optimal estimates for the fractional Hardy operator (2015)
  • 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
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Musielak-Orlicz-Sobolev spaces on metric measure spaces (2015)
  • 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
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Boundedness of maximal operators and Sobolev's theorem for non-homogeneous central Morrey spaces of variable exponent (2015)
  • Author(s): Y. Mizuta, T. Ohno, T. Shimomura
  • Journal Title: Hokkaido Math. J. Volume: 44 Pages: 185-201
  • 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 spaces (2015)
  • 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
  • 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
• [Presentation] Boundary behavior of monotone Sobolev functions on John domains (2017)
  • Author(s): 下村 哲
  • Organizer: 第６０回関数論シンポジウム
  • Invited
• [Presentation] Sobolev inequalities for Musielak-Orlicz spaces (2017)
  • Author(s): Tetsu Shimomura
  • Organizer: Nonstandard growth phenomena 2017
  • Int'l Joint Research
• [Presentation] Optimal estimates for the fractional Hardy operator (2016)
  • 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