Atomic Decompositions of Localized Hardy Spaces with Variable Exponents and Applications



In this paper, we introduce the localized Hardy spaces with variable exponents \(h^{p(\cdot )}\) and establish a new atomic decomposition theorem for \(h^{p(\cdot )}\) by using the discrete Littlewood–Paley–Stein theory. As an application of atomic decomposition, we investigate molecule decomposition for \(h^{p(\cdot )}\). Moreover, pseudo-differential operators of order zero are shown to be bounded on \(h^{p(\cdot )}\).


Atomic decomposition Localized Hardy space Variable exponent analysis Pseudo-differential operator Littlewood–Paley–Stein square function 

Mathematics Subject Classification

Primary 42B30 Secondary 42B25 42B35 46E30 



The project is sponsored by NUPTSF (Grant No.NY217151). The author also wishes to express his heartfelt thanks to the anonymous reviewer for corrections and so valuable suggestions.


  1. 1.
    Coifman, R.: A real variable characterization of \(H^p\). Studia Math. 51, 269–274 (1974)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Coifman, R., Meyer, Y.: Wavelets, Calderón-Zygmund and multilinear operators. Cambridge University Press, Cambridge (1992)MATHGoogle Scholar
  3. 3.
    Cruz-Uribe, D., Fiorenza, A.: Variable Lebesgue Spaces: Foundations and Harmonic Analysis, Birkhäuser (Basel, 2013)Google Scholar
  4. 4.
    Cruz-Uribe, D., Fiorenza, A., Martell, J., Pérez, C.: The boundedness of classical operators on variable \(L^p\) spaces. Ann. Acad. Sci. Fenn. Math. 31, 239–264 (2006)MathSciNetMATHGoogle Scholar
  5. 5.
    Cruz-Uribe, D., Wang, L.: Variable Hardy spaces. Indiana Univ. Math. J. 63, 447–493 (2014)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Deng, D., Han, Y-S.: Harmonic analysis on spaces of homogeneous type. Lecture Notes in Mathematics, 1966. Springer-Verlag, Berlin (2009)Google Scholar
  7. 7.
    Diening, L., Harjulehto, P., Hästö, P., Růžička, M.: Lebesgue and Sobolev spaces with variable exponents. Springer, Heidelberg (2011)CrossRefMATHGoogle Scholar
  8. 8.
    Frazier, M., Jawerth, B.: A discrete transform and decompositions of distribution spaces. J. Funct. Anal. 93(1), 34–170 (1990)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Frazier, M., Jawerth, B., Weiss, G.: Littlewood-Paley theory and the study of function spaces. In: CBMS Regional Conference Series in Mathematics, vol. 79. Published for the Conference Board of the Mathematical Sciences, Washington, DC. American Mathematical Society, Providence, RI (1991)Google Scholar
  10. 10.
    Goldberg, D.: A local version of real Hardy spaces. Duke Math. J. 46(1), 27–42 (1979)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Han, Y-S, Sawyer, E.T.: Littlewood-Paley Theory on Spaces of Homogeneous Type and the Classical Function Spaces. Mem. Amer. Math. Soc. 110, no. 530 (1994)Google Scholar
  12. 12.
    Ho, K.: Atomic decompositions of weighted Hardy spaces with variable exponents. Tohoku Math. J. (2) 69(3), 383–413 (2017)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Kováčik, O., Rákosník, J.: On spaces \(L^{p(x)}\) and \(W^{k, p(x)}\). Czechoslovak Math. J. 41, 592–618 (1991)MathSciNetMATHGoogle Scholar
  14. 14.
    Latter, R.: A characterization of \(H^p(\mathbb{R}^n)\) in terms of atoms. Studia Math. 62(1), 93–101 (1978)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Meyer, Y.: Wavelets and Operators. Cambridge University Press, Cambridge (1992)MATHGoogle Scholar
  16. 16.
    Nakai, E., Sawano, Y.: Hardy spaces with variable exponents and generalized Campanato spaces. J. Funct. Anal. 262, 3665–3748 (2012)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Pick, L., Růžičkap, M.: An example of a space \(L^{p(x)}\) on which the Hardy-Littlewood maximal operator is not bounded. Expos. Math. 19, 369–371 (2001)CrossRefMATHGoogle Scholar
  18. 18.
    Sawano, Y.: Atomic decompositions of Hardy spaces with variable exponents and its application to bounded linear operators. Integr. Equ. Oper. Theory 77, 123–148 (2013)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Tan, J.: Atomic decomposition of variable Hardy spaces via Littlewood-Paley-Stein theory. Ann. Funct. Anal. 9(1), 87–100 (2018)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Tan, J., Zhao, J.: Fractional integrals on variable Hardy-Morrey spaces. Acta. Math. Hung. 148(1), 174–190 (2016)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Triebel, H.: Theory of Function Spaces. Birkhäuser-Verlag, Basel (1983)CrossRefMATHGoogle Scholar
  22. 22.
    Yang, D., Zhuo, C., Nakai, E.: Characterizations of variable exponent Hardy spaces via Riesz transforms. Rev. Mat. Complut. 29(2), 245–270 (2016)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Zhuo, C., Yang, D.: Maximal function characterizations of variable Hardy spaces associated with non-negative self-adjoint operators satisfying Gaussian estimates. Nonlinear Anal. 141, 16–42 (2016)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Zhuo, C., Yang, D., Liang, Y.: Intrinsic square function characterizations of Hardy spaces with variable exponents. Bull. Malays. Math. Sci. Soc. 2(4), 1541–1577 (2016)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.College of ScienceNanjing University of Posts and TelecommunicationsNanjingChina

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