Advertisement

Revista Matemática Complutense

, Volume 25, Issue 2, pp 413–434 | Cite as

Maximal functions, Riesz potentials and Sobolev embeddings on Musielak-Orlicz-Morrey spaces of variable exponent in \({\bf R}^{n}\)

  • Yoshihiro Mizuta
  • Eiichi Nakai
  • Takao Ohno
  • Tetsu Shimomura
Article

Abstract

Our aim in this paper is to deal with Sobolev embeddings for Riesz potentials of variable order with functions in variable exponent Musielak-Orlicz-Morrey spaces.

Keywords

Maximal functions Musielak-Orlicz-Morrey space of variable exponent Riesz potential Sobolev embeddings Sobolev’s inequality 

Mathematics Subject Classification (2000)

31B15 46E30 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Acerbi, E., Mingione, G.: Regularity results for stationary electro-rheological fluids. Arch. Ration. Mech. Anal. 164, 213–259 (2002) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Acerbi, E., Mingione, G.: Gradient estimates for the p(x)-Laplacian system. J. Reine Angew. Math. 584, 117–148 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Adams, D.R.: A note on Riesz potentials. Duke Math. J. 42, 765–778 (1975) MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Almeida, A., Hasanov, J., Samko, S.: Maximal and potential operators in variable exponent Morrey spaces. Georgian Math. J. 15, 195–208 (2008) MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bendahmane, M., Wittbold, P.: Renormalized solutions for nonlinear elliptic equations with variable exponents and L 1 data. Nonlinear Anal. 70(2), 567–583 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Capone, C., Cruz-Uribe, D., Fiorenza, A.: The fractional maximal operator on variable L p spaces. Rev. Mat. Iberoam. 23(3), 743–770 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Chen, Y., Levine, S., Rao, M.: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math. 66(4), 1383–1406 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Chiarenza, F., Frasca, M.: Morrey spaces and Hardy-Littlewood maximal function. Rend. Mat. Appl. 7(7), 273–279 (1987) MathSciNetzbMATHGoogle Scholar
  9. 9.
    Cruz-Uribe, D., Fiorenza, A.: Llog L results for the maximal operator in variable L p spaces. Trans. Am. Math. Soc. 361, 2631–2647 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Cruz-Uribe, D., Fiorenza, A., Martell, J.M., Pérez, C.: The boundedness of classical operators in variable L p spaces. Ann. Acad. Sci. Fenn. Math. 31(1), 239–264 (2006) MathSciNetGoogle Scholar
  11. 11.
    Cruz-Uribe, D., Fiorenza, A., Neugebauer, C.J.: The maximal function on variable L p spaces. Ann. Acad. Sci. Fenn. Math. 28, 223–238 (2003). Ann. Acad. Sci. Fenn. Math. 29, 247–249 (2004) MathSciNetzbMATHGoogle Scholar
  12. 12.
    Dai, G.: Infinitely many solutions for a p(x)-Laplacian equation in \({\bf R}^{N}\). Nonlinear Anal. 71(3–4), 1133–1139 (2009) MathSciNetzbMATHGoogle Scholar
  13. 13.
    Diening, L.: Maximal functions in generalized L p(⋅) spaces. Math. Inequal. Appl. 7(2), 245–254 (2004) MathSciNetzbMATHGoogle Scholar
  14. 14.
    Diening, L.: Riesz potentials and Sobolev embeddings on generalized Lebesgue and Sobolev spaces L p(⋅) and W k,p(⋅). Math. Nachr. 263(1), 31–43 (2004) MathSciNetCrossRefGoogle Scholar
  15. 15.
    Diening, L., Hästö, P., Nekvinda, A.: Open problems in variable exponent Lebesgue and Sobolev spaces. In: Drabek, P., Rakosnik, J. (eds.) FSDONA04 Proceedings, pp. 38–58. Milovy, Czech Republic (2004) Google Scholar
  16. 16.
    Futamura, T., Mizuta, Y., Shimomura, T.: Sobolev embeddings for Riesz potential space of variable exponent. Math. Nachr. 279(13–14), 1463–1473 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Guliyev, V.S., Hasanov, J., Samko, S.: Boundedness of the maximal, potential and singular operators in the generalized variable exponent Morrey spaces. Math. Scand. 107, 285–304 (2010) MathSciNetzbMATHGoogle Scholar
  18. 18.
    Guliyev, V.S., Hasanov, J., Samko, S.: Boundedness of the maximal, potential and singular integral operators in the generalized variable exponent Morrey type spaces, Journal of Mathematical. Sciences 170(4), 423–443 (2010) MathSciNetGoogle Scholar
  19. 19.
    Harjulehto, P., Hästö, P., Latvala, V.: Sobolev embeddings in metric measure spaces with variable dimension. Math. Z. 254(3), 591–609 (2006) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Harjulehto, P., Hästö, P., Lê, U.V., Nuortio, M.: Overview of differential equations with non-standard growth. Nonlinear Anal. 72(12), 4551–4574 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Harjulehto, P., Hästö, P., Mizuta, Y., Shimomura, T.: Iterated maximal functions in variable exponent Lebesgue spaces. Manusucr. Math., to appear Google Scholar
  22. 22.
    Hästö, P.: Local-to-global results in variable exponent spaces. Math. Res. Lett. 16(2), 263–278 (2009) MathSciNetzbMATHGoogle Scholar
  23. 23.
    Hedberg, L.I.: On certain convolution inequalities. Proc. Am. Math. Soc. 36, 505–510 (1972) MathSciNetCrossRefGoogle Scholar
  24. 24.
    Kokilashvili, V., Samko, S.: Maximal and fractional operators in weighted L p(x) spaces. Rev. Mat. Iberoam. 20(2), 493–515 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Maeda, F.-Y., Mizuta, Y., Ohno, T.: Approximate identities and Young type inequalities in variable Lebesgue-Orlicz spaces L p(⋅)(log L)q(⋅). Ann. Acad. Sci. Fenn. Math. 35, 405–420 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Mizuta, Y., Nakai, E., Ohno, T., Shimomura, T.: An elementary proof of Sobolev embeddings for Riesz potentials of functions in Morrey spaces L 1,ν,β(G). Hiroshima Math. J. 38, 461–472 (2008) MathSciNetGoogle Scholar
  27. 27.
    Mizuta, Y., Nakai, E., Ohno, T., Shimomura, T.: Boundedness of fractional integral operators on Morrey spaces and Sobolev embeddings for generalized Riesz potentials. J. Math. Soc. Jpn. 62, 707–744 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Mizuta, Y., Nakai, E., Ohno, T., Shimomura, T.: Riesz potentials and Sobolev embeddings on Morrey spaces of variable exponent. Complex Var. Elliptic Equ., to appear Google Scholar
  29. 29.
    Mizuta, Y., Nakai, E., Ohno, T., Shimomura, T.: Sobolev’s inequality for Riesz potentials in Orlicz-Musielak spaces of variable exponent. Banach Funct. Spaces III, to appear Google Scholar
  30. 30.
    Mizuta, Y., Ohno, T., Shimomura, T.: Integrability of maximal functions for generalized Lebesgue spaces L p(⋅)(log L)q(⋅). Potential Theory Stoch. in Albac, 193–202 (2007) Google Scholar
  31. 31.
    Mizuta, Y., Ohno, T., Shimomura, T.: Integrability of maximal functions for generalized Lebesgue spaces with variable exponent. Math. Nachr. 281(3), 386–395 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Mizuta, Y., Ohno, T., Shimomura, T.: Sobolev’s inequalities and vanishing integrability for Riesz potentials of functions in the generalized Lebesgue space L p(⋅)(log L)q(⋅). J. Math. Anal. Appl. 345, 70–85 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Mizuta, Y., Shimomura, T.: Sobolev embeddings for Riesz potentials of functions in Morrey spaces of variable exponent. J. Math. Soc. Jpn. 60, 583–602 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Mizuta, Y., Shimomura, T.: Continuity properties of Riesz potentials of Orlicz functions. Tohoku Math. J. 61, 225–240 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Mizuta, Y., Shimomura, T.: Continuity properties for Riesz potentials of functions in Morrey spaces of variable exponent. Math. Inequal. Appl. 13, 99–122 (2010) MathSciNetzbMATHGoogle Scholar
  36. 36.
    Morrey, C.B.: On the solutions of quasi-linear elliptic partial differential equations. Trans. Am. Math. Soc. 43, 126–166 (1938) MathSciNetCrossRefGoogle Scholar
  37. 37.
    Musielak, J.: Orlicz Spaces and Modular Spaces. Lecture Notes in Math., vol. 1034. Springer, Berlin (1983) zbMATHGoogle Scholar
  38. 38.
    Nakai, E.: Hardy-Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces. Math. Nachr. 166, 95–103 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Nakai, E.: Generalized fractional integrals on Orlicz-Morrey spaces. In: Banach and Function Spaces, pp. 323–333. Yokohama Publ., Yokohama (2004) Google Scholar
  40. 40.
    Nakai, E.: Calderón-Zygmund operators on Orlicz-Morrey spaces and modular inequalities. In: Banach and Function Spaces II, pp. 393–410. Yokohama Publ., Yokohama (2008) Google Scholar
  41. 41.
    Nakai, E.: Orlicz-Morrey spaces and the Hardy-Littlewood maximal function. Stud. Math. 188, 193–221 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    O’Neil, R.: Fractional integration in Orlicz spaces. I. Trans. Am. Math. Soc. 115, 300–328 (1965) MathSciNetzbMATHGoogle Scholar
  43. 43.
    Peetre, J.: On the theory of L p,λ spaces. J. Funct. Anal. 4, 71–87 (1969) MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Růžička, M.: Electrorheological Fluids: Modeling and Mathematical Theory. Springer, Berlin (2000) zbMATHGoogle Scholar
  45. 45.
    Samko, S.: On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators. Integral Transforms Spec. Funct. 16(5–6), 461–482 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Serrin, J.: A remark on Morrey potential. Contemp. Math. 426, 307–315 (2007) MathSciNetCrossRefGoogle Scholar
  47. 47.
    Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Univ. Press, Princeton (1970) zbMATHGoogle Scholar

Copyright information

© Revista Matemática Complutense 2011

Authors and Affiliations

  • Yoshihiro Mizuta
    • 1
  • Eiichi Nakai
    • 2
  • Takao Ohno
    • 3
  • Tetsu Shimomura
    • 4
  1. 1.Department of Mathematics, Graduate School of ScienceHiroshima UniversityHigashi-HiroshimaJapan
  2. 2.Department of MathematicsIbaraki UniversityMito, IbarakiJapan
  3. 3.Faculty of Education and Welfare ScienceOita UniversityDannoharu Oita-cityJapan
  4. 4.Department of Mathematics, Graduate School of EducationHiroshima UniversityHigashi-HiroshimaJapan

Personalised recommendations