Skip to main content
SpringerLink
Log in

New characterizations of Musielak-Orlicz-Sobolev spaces via sharp ball averaging functions

  • Research Article
  • Published:
Frontiers of Mathematics in China Aims and scope Submit manuscript

Abstract

We establish a new characterization of the Musielak-Orlicz-Sobolev space on ℝn, which includes the classical Orlicz-Sobolev space, the weighted Sobolev space, and the variable exponent Sobolev space as special cases, in terms of sharp ball averaging functions. Even in a special case, namely, the variable exponent Sobolev space, the obtained result in this article improves the corresponding result obtained by P. Hästö and A. M. Ribeiro [Commun. Contemp. Math., 2017, 19: 1650022] via weakening the assumption f ∈ L1(ℝn) into f ∈ L1loc(ℝn), which was conjectured to be true by Hästö and Ribeiro in the aforementioned same article.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Acerbi E, Mingione G. Regularity results for a class of functionals with non-standard growth. Arch Ration Mech Anal, 2001, 156: 121–140

    Article  MathSciNet  MATH  Google Scholar 

  2. Acerbi E, Mingione G. Gradient estimates for the p(x)-Laplacean system. J Reine Angew Math, 2005, 584: 117–148

    Article  MathSciNet  MATH  Google Scholar 

  3. Adams R A, Fournier J J F. Sobolev Spaces. 2nd ed. Pure Appl Math, Vol 140. Amsterdam: Elsevier/Academic Press, 2003

    Google Scholar 

  4. Ahmida Y, Chlebicka I, Gwiazda P, Youssfi A. Gossez's approximation theorems in the Musielak-Orlicz-Sobolev spaces. J Funct Anal, 2018, 275: 2538–2571

    Article  MathSciNet  MATH  Google Scholar 

  5. Birnbaum Z, Orlicz W. Über die Verallgemeinerung des Begriffes der zueinander konjugierten Potenzen. Studia Math, 1931, 3: 1–67

    Article  MATH  Google Scholar 

  6. Bourgain J, Brezis H, Mironescu P. Another look at Sobolev spaces. In: Optimal Control and Partial Differential Equations. Amsterdam: IOS, 2001, 439–455

    Google Scholar 

  7. Bourgain J, Brezis H, Mironescu P. Limiting embedding theorems for W s,p when s ↑ 1 and applications. J Anal Math, 2002, 87: 77–101

    Article  MathSciNet  MATH  Google Scholar 

  8. Brezis H. How to recognize constant functions. A connection with Sobolev spaces. Russian Math Surveys, 2002, 57: 693–708

    Article  MathSciNet  MATH  Google Scholar 

  9. Colombo M, Mingione G. Bounded minimisers of double phase variational integrals. Arch Ration Mech Anal, 2015, 218: 219–273

    Article  MathSciNet  MATH  Google Scholar 

  10. Colombo M, Mingione G. Calderón-Zygmund estimates and non-uniformly elliptic operators. J Funct Anal, 2016, 270: 1416–1478

    Article  MathSciNet  MATH  Google Scholar 

  11. Cruz-Uribe D V, Fiorenza A. Variable Lebesgue Spaces. Foundations and Harmonic Analysis. Appl Numer Harmon Anal. Heidelberg: Birkhäuser/Springer, 2013

    MATH  Google Scholar 

  12. Diening L. Maximal function on generalized Lebesgue spaces L p(∙). Math Inequal Appl, 2004, 7: 245–253

    MathSciNet  MATH  Google Scholar 

  13. Diening L. Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces. Bull Sci Math, 2005, 129: 657–700

    Article  MathSciNet  MATH  Google Scholar 

  14. Diening L, Harjulehto P, Hästö P, Mizuta Y, Shimomura T. Maximal functions in variable exponent spaces: limiting cases of the exponent. Ann Acad Sci Fenn Math, 2009, 34: 503–522

    MathSciNet  MATH  Google Scholar 

  15. Diening L, Harjulehto P, Hästö P, Růžička M. Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Math, Vol 2017. Heidelberg: Springer, 2011

    MATH  Google Scholar 

  16. Diening L, Hästö P. Variable exponent trace spaces. Studia Math, 2007, 183: 127–141

    Article  MathSciNet  MATH  Google Scholar 

  17. Fernández Bonder J, Salort A M. Fractional order Orlicz-Sobolev spaces. arXiv: 1707.03267

  18. Ferreira R, Hästö P, Ribeiro A M. Characterization of generalized Orlicz spaces. Commun Contemp Math, 2018, https://doi.org/10.1142/S0219199718500797

    Google Scholar 

  19. Fiorenza A. A mean continuity type result for certain Sobolev spaces with variable exponent. Commun Contemp Math, 2002, 4: 587–605

    Article  MathSciNet  MATH  Google Scholar 

  20. Gallardo D. Orlicz spaces for which the Hardy-Littlewood maximal operator is bounded. Publ Mat, 1988, 32: 261–266

    Article  MathSciNet  MATH  Google Scholar 

  21. Grafakos L. Classical Fourier Analysis. 3rd ed. Graduate Texts in Math, Vol 249. New York: Springer, 2014

    Google Scholar 

  22. Harjulehto P, Hästö P, Klén R. Generalized Orlicz spaces and related PDE. Nonlinear Anal, 2016, 143: 155–173

    Article  MathSciNet  MATH  Google Scholar 

  23. Harjulehto P, Hästö P, Toivanen O. Hölder regularity of quasiminimizers under generalized growth conditions. Calc Var Partial Differential Equations, 2017, 56(2). Art 22 (26pp)

    Article  MATH  Google Scholar 

  24. Hästö P. The maximal operator on generalized Orlicz spaces. J Funct Anal, 2015, 269: 4038–4048

    Article  MathSciNet  MATH  Google Scholar 

  25. Hästö P, Ribeiro A M. Characterization of the variable exponent Sobolev norm without derivatives. Commun Contemp Math, 2017, 19: 1650022 (13pp)

    Article  MathSciNet  MATH  Google Scholar 

  26. Kokilashvili V, Krbec M. Weighted Inequalities in Lorentz and Orlicz Spaces. River Edge: World Scientific Publishing Co Inc, 1991

    Book  MATH  Google Scholar 

  27. Krasnosel'skií M A, Rutickií Ja B. Convex Functions and Orlicz Spaces. Groningen: P Noordhoff Ltd, 1961

    Google Scholar 

  28. Musielak J. Orlicz Spaces and Modular Spaces. Lecture Notes in Math, Vol 1034. Berlin: Springer-Verlag, 1983

    Google Scholar 

  29. Nakano H. Topology of Linear Topological Spaces. Tokyo: Maruzen Co Ltd, 1951

    Google Scholar 

  30. Nakano H. Modulared Semi-Ordered Linear Spaces. Tokyo: Maruzen Co Ltd, 1951

    MATH  Google Scholar 

  31. Ohno T, Shimomura T. Musielak-Orlicz-Sobolev spaces on metric measure spaces. Czechoslovak Math J, 2015, 65(140): 435–474

    Article  MathSciNet  MATH  Google Scholar 

  32. Ohno T, Shimomura T. Musielak-Orlicz-Sobolev spaces with zero boundary values on metric measure spaces. Czechoslovak Math J, 2016, 66(141): 371–394

    Article  MathSciNet  MATH  Google Scholar 

  33. Orlicz W. Über eine gewisse Klasse von Räumen vom Typus B. Bull Int Acad Pol Ser A, 1932, 8: 207–220

    MATH  Google Scholar 

  34. Rao M M, Ren Z. Theory of Orlicz Spaces. New York: Marcel Dekker, 1991

    MATH  Google Scholar 

  35. Rao M M, Ren Z. Applications of Orlicz Spaces. New York: Marcel Dekker, 2002

    MATH  Google Scholar 

  36. Squassina M, Volzone B. Bourgain-Brézis-Mironescu formula for magnetic operators. C R Math Acad Sci Paris, 2016, 354: 825–831

    Article  MathSciNet  MATH  Google Scholar 

  37. Turesson B O. Nonlinear Potential Theory and Weighted Sobolev Spaces. Lecture Notes in Math, Vol 1736. Berlin: Springer-Verlag, 2000

    Google Scholar 

  38. Yang D, Liang Y, Ky L D. Real-Variable Theory of Musielak-Orlicz Hardy Spaces. Lecture Notes in Math, Vol 2182. Cham: Springer, 2017

    Google Scholar 

  39. Youssfi A, Ahmida Y. Some approximation results in Musielak-Orlicz spaces. arXiv: 1708.02453

Download references

Acknowledgements

The authors would like to thank both referees for their very careful reading and several valuable comments which indeed improve the presentation of this article. This work was supported by the National Natural Science Foundation of China (Grant Nos. 11871254, 11571289, 11571039, 11761131002, 11671185, 11871100) and the Fundamental Research Funds for the Central Universities (Grant No. lzujbky-2018-111).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Gansu Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, 730000, China

    Sibei Yang

  2. Laboratory of Mathematics and Complex Systems (Ministry of Education of China), School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China

    Dachun Yang & Wen Yuan

Authors
  1. Sibei Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dachun Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Wen Yuan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dachun Yang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, S., Yang, D. & Yuan, W. New characterizations of Musielak-Orlicz-Sobolev spaces via sharp ball averaging functions. Front. Math. China 14, 177–201 (2019). https://doi.org/10.1007/s11464-019-0744-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11464-019-0744-1

Keywords

MSC

Search

Navigation