Skip to main content
SpringerLink
Log in

Besov spaces with variable smoothness and integrability on Lie groups of polynomial growth

  • Published:
Journal of Pseudo-Differential Operators and Applications Aims and scope Submit manuscript

Abstract

In this paper, we consider the Besov spaces with variable smoothness and integrability on Lie groups of polynomial growth. We prove that such function spaces are well defined in the sense that their definitions are independent of the choice of basis functions under some specific assumptions. Then we show some embeddings related to them.

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. Adamowicz, T., Harjulehto, P., Hästö, P.: Maximal operator in variable exponent Lebesgue spaces on unbounded quasimetric measure spaces. Math. Scand. 116(1), 5–22 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alexopoulos, G.: Spectral multipliers on Lie groups of polynomial growth. Proc. Am. Math. Soc. 120(3), 973–979 (1994)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Besov, O.V.: Embeddings of spaces of differentiable functions of variable smoothness. Tr. Mat. Inst. Steklova 214(3), 25–58 (1997)

    MathSciNet  MATH  Google Scholar 

  5. Besov, O.V.: On spaces of functions of variable smoothness defined by pseudodifferential operators. Tr. Mat. Inst. Steklova 227(4), 56–74 (1999)

    MathSciNet  MATH  Google Scholar 

  6. Besov, O.V.: Equivalent normings of spaces of functions of variable smoothness. Dokl. Akad. Nauk. 391(5), 583–586 (2003)

    MathSciNet  MATH  Google Scholar 

  7. Christ, M.: \(L^p\) bounds for spectral multipliers on nilpotent groups. Trans. Am. Math. Soc. 328(1), 73–81 (1991)

    MathSciNet  MATH  Google Scholar 

  8. Cruz–Uribe D.V., Fiorenza A.: Variable Lebesgue spaces. Birkhäuser, Basel (2013)

  9. Diening, L., Harjulehto, P., Hästö, P., Ruzicka M.: Lebesgue and Sobolev spaces with variable exponents. Springer, Berlin, Heidelberg (2011)

  10. Diening, L., Hästö, P., Roudenko, S.: Function spaces of variable smoothness and integrability. J. Funct. Anal. 256(6), 1731–1768 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dungey, N., Elst, A.F.M.T., Robinson, D.W.: Analysis on Lie Groups with Polynomial Growth. Birkhäuser Boston (2003)

  12. Ehrenpreis, L.: Some properties of distributions on Lie groups. Pac. J. Math. 6(4), 591–605 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  13. Furioli, G., Melzi, C., Veneruso, A.: Littlewood–Paley decompositions and Besov spaces on Lie groups of polynomial growth. Math. Nachr. 279(9–10), 1028–1040 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Grafakos, L.: Classical Fourier Analysis. Graduate Texts in Mathematics. Springer, New York (2008)

    MATH  Google Scholar 

  15. Grafakos, L.: Modern Fourier Analysis. Graduate Texts in Mathematics. Springer, New York (2008)

    MATH  Google Scholar 

  16. Musielak, J.: Orlicz Spaces and Modular Spaces. Lecture Notes in Mathematics. Springer, Berlin (1983)

    Book  MATH  Google Scholar 

  17. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. I. Functional Analysis, p. xv+400. Academic Press, London (1980)

    Google Scholar 

  18. Skrzypczak, L.: Besov spaces and Hausdorff dimension for some Carnot–Carathéodory metric spaces. Can. J. Math. 54(6), 1280–1304 (2002)

    Article  MATH  Google Scholar 

  19. Stein, E.M., Murphy, T.S.: Harmonic analysis: real–variable methods, orthogonality, and oscillatory integrals. Princeton University Press (1993)

  20. Stein, E.M., Weiss, G.: Introduction to Fourier analysis on Euclidean spaces. Princeton University Press (1971)

  21. Varopoulos, N.T., Saloff-Coste, L., Coulhon, T.: Analysis and Geometry on Groups. Cambridge University Press, Cambridge (1992)

    MATH  Google Scholar 

  22. Varopoulos, N.T.: Analysis on Lie groups. Rev. Mat. Iberoam. 12(3), 791–917 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  23. Xu, J.S.: Variable Besov and Triebel–Lizorkin spaces. Ann. Acad. Sci. Fenn. Math. 33(2), 511–522 (2008)

    MathSciNet  MATH  Google Scholar 

  24. Xu, J.S.: The relation between variable Bessel potential spaces and Triebel–Lizorkin spaces. Integr. Transf. Spec. Funct. 19(8), 599–605 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  25. Xu, J.S.: An atomic decomposition of variable Besov and Triebel–Lizorkin spaces. Armen. J. Math. 2(1), 1–12 (2009)

    MathSciNet  MATH  Google Scholar 

  26. Yang, D.C., Zhuo, C.Q., Yuan, W.: Besov-type spaces with variable smoothness and integrability. J. Funct. Anal. 269(6), 1840–1898 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  27. Yang, D.C., Zhuo, C.Q., Yuan, W.: Triebel–Lizorkin-type spaces with variable exponents. Banach J. Math. Anal. 9(4), 146–202 (2015)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors would like to express great gratitude to the referees for the valuable comments and helpful suggestions.

Author information

Authors and Affiliations

  1. Key Laboratory of Mathematics and Complex Systems, Ministry of Education, Institution of Mathematics and Mathematical Education, School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China

    Jingxuan Fang & Jiman Zhao

Authors
  1. Jingxuan Fang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jiman Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jiman Zhao.

Additional information

Jiman Zhao: Supported by National Natural Science Foundation of China (Grant No. 11471040), National Natural Science Foundation of China (NSFC)—Deutsche Forschungsgemeinschaft (DFG) (No. 11761131002) and the Fundamental Research Funds for the Central Universities (No. 2014KJJCA10).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fang, J., Zhao, J. Besov spaces with variable smoothness and integrability on Lie groups of polynomial growth. J. Pseudo-Differ. Oper. Appl. 10, 581–599 (2019). https://doi.org/10.1007/s11868-018-0246-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11868-018-0246-z

Keywords

Mathematics Subject Classification

Search

Navigation