Potential theoretic study for elliptic partial differential equations

• Project/Area Number: 24540174
• 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(研究院), 教授 (50294476)
• Co-Investigator(Kenkyū-buntansha): MIZUTA Yoshihiro 広島工業大学, 工学部, 教授 (00093815)
• Project Period (FY): 2012-04-01 – 2015-03-31
• Project Status: Completed (Fiscal Year 2014)
• Budget Amount: ¥5,070,000 (Direct Cost: ¥3,900,000、Indirect Cost: ¥1,170,000); Fiscal Year 2014: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2013: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2012: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000)
• 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 with variable exponents. As an application of the boundedness of the maximal operator, we establish a generalization of Sobolev's inequality and Trudinger's exponential inequality for Riesz potentials of functions in Musielak-Orlicz-Morrey spaces with variable exponents.

Research Products

• [Journal Article] Trudinger's inequality for Riesz potentialsof functions in Musielak-Orlicz spaces (2014)
  • Author(s): Takao Ohno and Tetsu Shimomura
  • Journal Title: Bull. Sci. Math Volume: 138 Issue: 2 Pages: 225-235
  • DOI: 10.1016/j.bulsci.2013.05.007
  • Peer Reviewed
• [Journal Article] Sobolev embeddings for Riesz potentials of functions in grand Morrey spaces of variable exponents over non-doubling measure spaces (2014)
  • Author(s): T. Ohno, T. Shimomura
  • Journal Title: Czechoslovak Math. J. Volume: 64 Issue: 1 Pages: 209-228
  • DOI: 10.1007/s10587-014-0095-8
  • Peer Reviewed
• [Journal Article] Hardy's inequality in Musielak-Orlicz-Sobolev spaces (2014)
  • Author(s): F. - Y. Maeda, Y. Mizuta, T. Ohno, T. Shimomura
  • Journal Title: Hiroshima Math. J. Volume: 44 Pages: 139-155
  • Peer Reviewed
• [Journal Article] Trudinger's inequality and continuity for Riesz potentials of functions in Musielak-Orlicz-Morrey spaces on metric measure spaces (2014)
  • Author(s): T. Ohno, T. Shimomura
  • Journal Title: Nonlinear Anal. Volume: 106 Pages: 1-17
  • DOI: 10.1016/j.na.2014.04.008
  • Peer Reviewed
• [Journal Article] Sobolev embeddings for Riesz potentials of functions in Musielak-Orlicz-Morrey spaces over non-doubling measure spaces (2014)
  • Author(s): Y. Sawano, T. Shimomura
  • Journal Title: Integral Transforms and Special Functions Volume: 25 Issue: 12 Pages: 976-991
  • DOI: 10.1080/10652469.2014.955099
  • Peer Reviewed
• [Journal Article] Sobolev embeddings for Riesz potentials of functions in Morrey spaces $L^{1,\nu,\beta_1,\beta_2}(R^n)$ (2014)
  • Author(s): Y. Mizuta, T. Shimomura
  • Journal Title: Math. Inequal. Appl. Volume: 17 Issue: 4 Pages: 1399-1414
  • DOI: 10.7153/mia-17-103
  • Peer Reviewed
• [Journal Article] Boundedness of maximal operators and Sobolev's inequality on Musielak-Orlicz-Morrey spaces (2013)
  • Author(s): Fumi-Yuki Maeda, Yoshihiro Mizuta, Takao Ohno and Tetsu Shimomura
  • Journal Title: Bull. Sci. Math Volume: 137 Issue: 1 Pages: 76-96
  • DOI: 10.1016/j.bulsci.2012.03.008
  • Peer Reviewed
• [Journal Article] Trudinger's inequality and continuity of potentials on Musielak-Orlicz-Morrey spaces (2013)
  • Author(s): Fumi-Yuki Maeda, Yoshihiro Mizuta, Takao Ohno and Tetsu Shimomura
  • Journal Title: Potential Anal Volume: 38 Issue: 2 Pages: 515-535
  • DOI: 10.1007/s11118-012-9284-y
  • Peer Reviewed
• [Journal Article] Sobolev embeddings for Riesz potentials of functions in non-doubling Morrey spaces of variable exponents (2013)
  • Author(s): Yoshihiro Sawano and Tetsu Shimomura
  • Journal Title: Collectanea Mathematica Volume: online, open access Issue: 3 Pages: 1-38
  • DOI: 10.1007/s13348-013-0082-7
  • Peer Reviewed
• [Journal Article] Approximate identities and Young type inequalities in Musielak-Orlicz spaces (2013)
  • Author(s): F. - Y. Maeda, Y. Mizuta, T. Ohno, T. Shimomura
  • Journal Title: Czechoslovak Math. J. Volume: 63 Issue: 4 Pages: 933-948
  • DOI: 10.1007/s10587-013-0063-8
  • Peer Reviewed
• [Journal Article] Sobolev's inequality for Riesz potentials of functions in generalized Morrey spaces with variable exponent attaining the value 1 over non-doubling measure spaces (2013)
  • Author(s): Y. Sawano, T. Shimomura
  • Journal Title: Journal of Inequalities and Applications Volume: 2013 Issue: 1
  • DOI: 10.1186/1029-242x-2013-12
  • Peer Reviewed
• [Journal Article] Littlewood-Paley theory for variable exponent Lebesgue spaces and Gagliardo-Nirenberg inequality for Riesz potentials (2013)
  • Author(s): Yoshihiro Mizuta, Eiichi Nakai, Yoshihiro Sawano and Tetsu Shimomura
  • Journal Title: Journal of the Mathematical Society of Japan Volume: 65 Issue: 2 Pages: 633-670
  • DOI: 10.2969/jmsj/06520633
  • NAID: 10031177293
  • ISSN: 0025-5645, 1881-1167, 1881-2333
  • Peer Reviewed
• [Journal Article] Weighted Sobolev inequality in Musielak-Orlicz space (2012)
  • Author(s): Y. Mizuta, T. Shimomura
  • Journal Title: J. Math. Anal. Appl. Volume: 388 Pages: 86-97
  • Peer Reviewed
• [Journal Article] Gagliardo-Nirenberg inequality for generalized Riesz potentials of functions in Musielak-Orlicz spaces (2012)
  • Author(s): Yoshihiro Mizuta, Eiichi Nakai, Yoshihiro Sawano, Tetsu Shimomura
  • Journal Title: Archiv der Mathematik Volume: 98 Issue: 3 Pages: 253-263
  • DOI: 10.1007/s00013-012-0362-6
  • Peer Reviewed
• [Journal Article] Maximal functions, Riesz potentials and Sobolev embeddings on Musielak-Orlicz-Morrey spaces of variable exponent in R^n (2012)
  • Author(s): Yoshihiro Mizuta
  • Journal Title: Rev. Mat. Complut. Volume: 25 Issue: 2 Pages: 413-434
  • DOI: 10.1007/s13163-011-0074-7
  • Peer Reviewed
• [Journal Article] A trace inequality of Riesz potentials in variable exponent Orlicz spaces (2012)
  • Author(s): Y. Mizuta, T. Shimomura
  • Journal Title: Math. Nachr. Volume: 285 Issue: 11-12 Pages: 1466-1485
  • DOI: 10.1002/mana.201000115
  • Peer Reviewed
• [Presentation] Growth properties of Musielak-Orlicz integral means for Riesz potentials (2014)
  • Author(s): 下村 哲
  • Organizer: 日本数学会秋季総合分科会
  • Place of Presentation: 広島大学
  • Year and Date: 2014-09-27
• [Presentation] Weighted Morrey spaces of variable exponent and Riesz potentials
  • Author(s): 下村 哲
  • Organizer: ポテンシャル論研究集会
  • Place of Presentation: 北海道大学
• [Presentation] Hardy averaging operator on generalized Banach function spaces
  • Author(s): 下村 哲
  • Organizer: RIMS workshop Potential Theory and its Related Fields
  • Place of Presentation: Kyoto University, Fuculty of Engineering
• [Presentation] Exponential integrability of Riesz potentials of Orlicz functions
  • Author(s): 下村 哲
  • Organizer: Harmonic Analysis and its Applications at Tokyo 2012
  • Place of Presentation: Tokyo Metropolitan University, Minami-Osawa Campus