Revista Matemática Complutense

, Volume 29, Issue 2, pp 341–404 | Cite as

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

  • Takahiro NoiEmail author


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 \({\mathbb R}^n_+\).


Besov space Triebel-Lizorkin space Variable exponents Quarkonial decomposition Trace operator Extension operator 

Mathematics Subject Classification

Primary 42B35 Secondary 41A17 



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.


  1. 1.
    Almeida, A., Hästö, P.: Besov spaces with variable smoothness and integrability. J. Funct. Anal. 258(5), 1628–1655 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Antontsev, S., Rodrigues, J.: On stationary thermo-rheological viscous flows. Ann. Univ. Ferrara Sez. VII. Sci. Mat. 52, 19–36 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 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)MathSciNetzbMATHGoogle Scholar
  4. 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)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Diening, L.: Maximal function on Musielak-Orilcz spaces and generalized lebesgue spaces Bull. Sci. Math. 129, 657–700 (2005)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Diening, L., Harjulehto, P., Hästö, P., Růžička, M.: Lebesgue and sobolev spaces with variable exponents, Lecture Notes in Mathematics. Springer, New York (2011)Google Scholar
  7. 7.
    Diening, L., Hästö, P., Roudenko, S.: Function spaces of variable smoothness and integrability. J. Funct. Anal. 256, 1731–1768 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Drihem, D.: Atomic decomposition of Besov spaces with variable smoothness and integrability. J. Math. Anal. Appl. 389, 15–31 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Frazier, M., Jawerth, B.: Decomposition of Besov spaces. Indiana Univ. Math. J. 34, 777–799 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Frazier, M., Jawerth, B.: A discrete transform and decompositions of distribution spaces. J. Funct. Anal. 93, 34–170 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Franke, J., Runst, T.: Regular elliptic boundary value problems in Besov-Triebel-Lizorkin space. Math. Nachr. 174, 113–149 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kempka, H.: 2-Microlocal Besov and Triebel-Lizorkin spaces of variable integrability. Rev. Mat. Complut. 22, 227–251 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 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)CrossRefzbMATHGoogle Scholar
  18. 18.
    Kováčik, O., Rákosník, J.: On spaces \(L^{p(x)}\) and \(W^{k, p(x)}\). Czech. Math. J. 41, 592–618 (1991)zbMATHGoogle Scholar
  19. 19.
    Kyriazis, G.: Decomposition systems for function spaces. Studia Math. 157, 133–169 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Moura, S., Neves, J., Schneider, C.: On trace spaces of 2-microlocal Besov spaces with variable integrability. Math. Nachr. 286, 1240–1254 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Nakamura, S., Noi T., Sawano, Y.: Generalized Morrey spaces and trace operator. Sci. China Math. (2016) (to appear)Google Scholar
  22. 22.
    Noi, T.: Fourier multiplier theorems for Besov and Triebel-Lizorkin spaces with variable exponents. Math. Inequal. Appl. 17, 49–74 (2014)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Noi, T., Izuki, M.: Duality of Besov, Triebel-Lizorkin and Herz spaces with variable exponents. Rend. Circ. Mat. Palermo. 63, 221–245 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Orlicz, W.: Über konjugierte exponentenfolgen. Studia Math. 3, 200–211 (1931)zbMATHGoogle Scholar
  26. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Rajagopal, K., Růžička, M.: On the modeling of electrorheological materials. Mech. Res. Comm. 23, 401–407 (1996)CrossRefzbMATHGoogle Scholar
  28. 28.
    Sawano, Y.: Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces on domains. Math. Nachr. 283(10), 1456–1487 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Sawano, Y.: Theory of Besov spaces. Nihon Hyo-ronsya (2011) (in Japanese)Google Scholar
  30. 30.
    Sawano, Y., Tanaka, H.: Decompositions of Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces. Math. Z. 257(4), 871–905 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Triebel, H.: Theory of function spaces. Birkhäuser, Basel, Boston (1983)CrossRefzbMATHGoogle Scholar
  32. 32.
    Triebel, H.: The structure of functions. Birkhäuser, Basel, Boston (2001)CrossRefzbMATHGoogle Scholar

Copyright information

© Universidad Complutense de Madrid 2016

Authors and Affiliations

  1. 1.Department of Mathematics and Information ScienceTokyo Metropolitan UniversityHachiojiJapan

Personalised recommendations