Skip to main content
SpringerLink
Log in

Trace and extension operators for Besov spaces and Triebel–Lizorkin spaces with variable exponents

  • Published:
Revista Matemática Complutense Aims and scope Submit manuscript

Abstract

This paper is concerned with the boundedness of trace and extension operators for Besov spaces and Triebel–Lizorkin spaces with variable exponents on the upper half space \({\mathbb R}^n_+\). To define trace and extension operators, we introduce a quarkonial decomposition for Besov spaces and Triebel–Lizorkin spaces with variable exponents on \({\mathbb R}^n\). We then study the continuity of such operators related to \({\mathbb R}^n_+\).

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. Almeida, A., Hästö, P.: Besov spaces with variable smoothness and integrability. J. Funct. Anal. 258(5), 1628–1655 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  2. Antontsev, S., Rodrigues, J.: On stationary thermo-rheological viscous flows. Ann. Univ. Ferrara Sez. VII. Sci. Mat. 52, 19–36 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cruz-Uribe, D., SFO, A., Fiorenza Martell, J.M., Pérez, C.: The boundedness of classical operators on variable \(L^p\) spaces. Ann. Acad. Sci. Fenn. Math. 31, 239–264 (2006)

    MathSciNet  MATH  Google Scholar 

  4. Diening, L., Růžička, M.: Calderón-Zygmund operators on generalized Lebesgue spaces \(L^{p(\cdot )}\) and problems related to fluid dynamics. J. Reine Angew. Math. 563, 197–220 (2003)

    MathSciNet  MATH  Google Scholar 

  5. Diening, L.: Maximal function on Musielak-Orilcz spaces and generalized lebesgue spaces Bull. Sci. Math. 129, 657–700 (2005)

    MathSciNet  MATH  Google Scholar 

  6. Diening, L., Harjulehto, P., Hästö, P., Růžička, M.: Lebesgue and sobolev spaces with variable exponents, Lecture Notes in Mathematics. Springer, New York (2011)

  7. Diening, L., Hästö, P., Roudenko, S.: Function spaces of variable smoothness and integrability. J. Funct. Anal. 256, 1731–1768 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Drihem, D.: Atomic decomposition of Besov spaces with variable smoothness and integrability. J. Math. Anal. Appl. 389, 15–31 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Frazier, M., Jawerth, B.: Decomposition of Besov spaces. Indiana Univ. Math. J. 34, 777–799 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  10. Frazier, M., Jawerth, B.: A discrete transform and decompositions of distribution spaces. J. Funct. Anal. 93, 34–170 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  11. Franke, J., Runst, T.: Regular elliptic boundary value problems in Besov-Triebel-Lizorkin space. Math. Nachr. 174, 113–149 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gonçalves, H.F., Moura, S.D., Neves, J.S.: On the trace spaces of 2-microlocal type spaces. J. Funct. Anal. 267, 3444–3468 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Huang, A., Xu, J.: Multilinear singular integrals and commutators in variable exponent Lebesgue spaces. Appl. Math. J. Chin. Univ. 25(1), 69–77 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kempka, H.: 2-Microlocal Besov and Triebel-Lizorkin spaces of variable integrability. Rev. Mat. Complut. 22, 227–251 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kempka, H.: Atomic, molecular and wavelet decomposition of 2-microlocal Besov and Triebel-Lizorkin spaces with variable integrability. Funct. Approx. Comment. Math. 43(2), 171–208 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kempka, H., Vybíral, J.: Spaces of variable smoothness and integrability: characterizations by local means and ball means of differences. J. Fourier Anal. Appl. 18, 852–891 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kempka, H., Vybíral, J.: A note on the spaces of variable integrability and summability of Almeida and Hästö. Proc. Am. Math. Soc. 141, 3207–3212 (2013)

    Article  MATH  Google Scholar 

  18. Kováčik, O., Rákosník, J.: On spaces \(L^{p(x)}\) and \(W^{k, p(x)}\). Czech. Math. J. 41, 592–618 (1991)

    MATH  Google Scholar 

  19. Kyriazis, G.: Decomposition systems for function spaces. Studia Math. 157, 133–169 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. Moura, S., Neves, J., Schneider, C.: On trace spaces of 2-microlocal Besov spaces with variable integrability. Math. Nachr. 286, 1240–1254 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Nakamura, S., Noi T., Sawano, Y.: Generalized Morrey spaces and trace operator. Sci. China Math. (2016) (to appear)

  22. Noi, T.: Fourier multiplier theorems for Besov and Triebel-Lizorkin spaces with variable exponents. Math. Inequal. Appl. 17, 49–74 (2014)

    MathSciNet  MATH  Google Scholar 

  23. Noi, T., Izuki, M.: Duality of Besov, Triebel-Lizorkin and Herz spaces with variable exponents. Rend. Circ. Mat. Palermo. 63, 221–245 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  24. Noi, T., Sawano, Y.: Complex interpolation of Besov spaces and Triebel-Lizorkin spaces with variable exponents. J. Math. Anal. Appl. 387, 676–690 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Orlicz, W.: Über konjugierte exponentenfolgen. Studia Math. 3, 200–211 (1931)

    MATH  Google Scholar 

  26. Pick, L., Růžička, M.: An example of a space \(L^{p(\cdot )}\) on which the Hardy-Littlewood maximal operator is not bounded. Expo. Math. 19, 369–371 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  27. Rajagopal, K., Růžička, M.: On the modeling of electrorheological materials. Mech. Res. Comm. 23, 401–407 (1996)

    Article  MATH  Google Scholar 

  28. Sawano, Y.: Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces on domains. Math. Nachr. 283(10), 1456–1487 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  29. Sawano, Y.: Theory of Besov spaces. Nihon Hyo-ronsya (2011) (in Japanese)

  30. Sawano, Y., Tanaka, H.: Decompositions of Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces. Math. Z. 257(4), 871–905 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  31. Triebel, H.: Theory of function spaces. Birkhäuser, Basel, Boston (1983)

    Book  MATH  Google Scholar 

  32. Triebel, H.: The structure of functions. Birkhäuser, Basel, Boston (2001)

    Book  MATH  Google Scholar 

Download references

Acknowledgments

The author would like to express my gratitude to Professor Yoshikazu Kobayashi for his reading this manuscript carefully and appropriate advice. The author also would like to express my gratitude to Professor Yoshihiro Sawano for sending his book [29]. The author obtained many ideas from the book [29]. The author is thankful to anonymous reviewers for their careful reading of this paper and their comments.

Author information

Authors and Affiliations

  1. Department of Mathematics and Information Science, Tokyo Metropolitan University, 1-1 Minami Osawa, Hachioji, Tokyo, 192-0397, Japan

    Takahiro Noi

Authors
  1. Takahiro Noi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Takahiro Noi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Noi, T. Trace and extension operators for Besov spaces and Triebel–Lizorkin spaces with variable exponents. Rev Mat Complut 29, 341–404 (2016). https://doi.org/10.1007/s13163-016-0191-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13163-016-0191-4

Keywords

Mathematics Subject Classification

Search

Navigation