• Wang, Hongbin (School of Science Shandong University of Technology)
  • Received : 2015.12.26
  • Published : 2017.05.01


Let ${\Omega}{\in}L^s(S^{n-1})$ for s > 1 be a homogeneous function of degree zero and b be BMO functions or Lipschitz functions. In this paper, we obtain some boundedness of the $Calder{\acute{o}}n$-Zygmund singular integral operator $T_{\Omega}$ and its commutator [b, $T_{\Omega}$] on Herz-type Hardy spaces with variable exponent.


Herz-type Hardy space;variable exponent;$Calder{\acute{o}}n$-Zygmund singular integral;commutator


  1. C. Capone, D. Cruz-Uribe, SFO and A. Fiorenza, The fractional maximal operator and fractional integrals on variable $L^p$ spaces, Rev. Mat. Iberoamericana 23 (2007), no. 3, 743-770.
  2. D. Cruz-Uribe and A. Fiorenza, Variable Lebesgue Spaces: Foundations and Harmonic Analysis (Applied and Numerical Harmonic Analysis), Springer, Heidelberg, 2013.
  3. D. Cruz-Uribe, A. Fiorenza, J. M. Martell, and C. Perez, The boundedness of classical operators on variable $L^p$ spaces, Ann. Acad. Sci. Fenn. Math. 31 (2006), no. 1, 239-264.
  4. L. Diening, P. Harjulehto, P. Hasto, and M. Ruzicka, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Math., vol. 2017, Springer, Heidelberg, 2011.
  5. M. Izuki, Boundedness of sublinear operators on Herz spaces with variable exponent and application to wavelet characterization, Anal. Math. 36 (2010), no. 1, 33-50.
  6. M. Izuki, Boundedness of commutators on Herz spaces with variable exponent, Rend. Circ. Mat. Palermo (2) 59 (2010), no. 2, 199-213.
  7. O. Kovacik and J. Rakosnik, On spaces $L^{p(x)}\;and\;W^{k,p(x)}$, Czechoslovak Math. J. 41 (1991), no. 4, 592-618.
  8. S. Lu, Y. Ding, and D. Yan, Singular Integrals and Related Topics, World Scientific Press, Beijing, 2011.
  9. S. Lu, Q. Wu, and D. Yang, Boundedness of commutators on Hardy type spaces, Sci. China Ser. A 45 (2002), no. 8, 984-997.
  10. E. Nakai and Y. Sawano, Hardy spaces with variable exponents and generalized Campanato spaces, J. Funct. Anal. 262 (2012), no. 9, 3665-3748.
  11. J. Tan and Z. Liu, Some boundedness of homogeneous fractional integrals on variable exponent function spaces, Acta Math. Sinica (Chin. Ser.) 58 (2015), no. 2, 309-320.
  12. H. Wang, The continuity of commutators on Herz-type Hardy spaces with variable exponent, Kyoto J. Math. 56 (2016), no. 3, 559-573.
  13. H. Wang, Boundedness of commutators on Herz-type Hardy spaces with variable exponent, Jordan J. Math. Stat. 9 (2016), no. 1, 17-30.
  14. H. Wang and Z. Liu, The Herz-type Hardy spaces with variable exponent and their applications, Taiwanese J. Math. 16 (2012), no. 4, 1363-1389.
  15. H. Wang and Z. Liu, Some characterizations of Herz-type Hardy spaces with variable exponent, Ann. Funct. Anal. 6 (2015), no. 2, 224-243.