Czechoslovak Mathematical Journal

, Volume 66, Issue 1, pp 251–269 | Cite as

Commutators of Marcinkiewicz integrals on Herz spaces with variable exponent

  • Hongbin WangEmail author


Let Ω ∈ L s (S n−1) for s ⩾ 1 be a homogeneous function of degree zero and b a BMO function. The commutator generated by the Marcinkiewicz integral \({\mu _\Omega }\) and b is defined by
$$\left[ {b,{\mu _\Omega }} \right](f)(x) = {\left( {\int_0^\infty {{{\left| {\int_{\left| {x - y} \right| \leqslant t} {\frac{{\Omega (x - y)}}{{{{\left| {x - y} \right|}^{n - 1}}}}\left[ {b(x) - b(y)} \right]f(y){\text{d}}y} } \right|}^2}\frac{{{\text{d}}t}}{{{t^3}}}} } \right)^{1/2}}$$
. In this paper, the author proves the (L p(·)(ℝ n ),L p(·)(ℝ n ))-boundedness of the Marcinkiewicz integral operator \({\mu _\Omega }\) and its commutator [b, \({\mu _\Omega }\)] when p(·) satisfies some conditions. Moreover, the author obtains the corresponding result about \({\mu _\Omega }\) and [b, \({\mu _\Omega }\)] on Herz spaces with variable exponent.


Herz space variable exponent commutator Marcinkiewicz integral 

MSC 2010

42B20 42B35 


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  1. [1]
    C. Capone, D. Cruz-Uribe, A. Fiorenza: The fractional maximal operator and fractional integrals on variable L p spaces. Rev. Mat. Iberoam. 23 (2007), 743–770.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    D. V. Cruz-Uribe, A. Fiorenza: Variable Lebesgue Spaces. Foundations and Harmonic Analysis. Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, New York, 2013.Google Scholar
  3. [3]
    D. Cruz-Uribe, A. Fiorenza, J. M. Martell, C. Perez: The boundedness of classical operators on variable L p spaces. Ann. Acad. Sci. Fenn. Math. 31 (2006), 239–264.MathSciNetzbMATHGoogle Scholar
  4. [4]
    L. Diening, P. Harjulehto, P. Hasto, M. Růžička: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics 2017, Springer, Berlin, 2011.zbMATHGoogle Scholar
  5. [5]
    Y. Ding, D. Fan, Y. Pan: Weighted boundedness for a class of rough Marcinkiewicz integrals. Indiana Univ. Math. J. 48 (1999), 1037–1055.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Y. Ding, S. Lu, K. Yabuta: On commutators of Marcinkiewicz integrals with rough kernel. J. Math. Anal. Appl. 275 (2002), 60–68.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    M. Izuki: Boundedness of commutators on Herz spaces with variable exponent. Rend. Circ. Mat. Palermo (2) 59 (2010), 199–213.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    M. Izuki: Boundedness of sublinear operators on Herz spaces with variable exponent and application to wavelet characterization. Anal. Math. 36 (2010), 33–50.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    O. Kovačik, J. Rakosnik: On spaces L p(x) and W k,p(x). Czech. Math. J. 41 (1991), 592–618.zbMATHGoogle Scholar
  10. [10]
    Z. Liu, H. Wang: Boundedness of Marcinkiewicz integrals on Herz spaces with variable exponent. Jordan J. Math. Stat. 5 (2012), 223–239.zbMATHGoogle Scholar
  11. [11]
    B. Muckenhoupt, R. L. Wheeden: Weighted norm inequalities for singular and fractional integrals. Trans. Am. Math. Soc. 161 (1971), 249–258.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    E. Nakai, Y. Sawano: Hardy spaces with variable exponents and generalized Campanato spaces. J. Funct. Anal. 262 (2012), 3665–3748.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    E. M. Stein: On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz. Trans. Am. Math. Soc. 88 (1958), 430–466; corr. ibid. 98 (1961), 186.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    J. Tan, Z. G. Liu: Some boundedness of homogeneous fractional integrals on variable exponent function spaces. Acta Math. Sin., Chin. Ser. 58 (2015), 309–320. (In Chinese. )MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    H. Wang, Z. Fu, Z. Liu: Higher-order commutators of Marcinkiewicz integrals on variable Lebesgue spaces. Acta Math. Sci., Ser. A Chin. Ed. 32 (2012), 1092–1101. (In Chinese. )MathSciNetzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2016

Authors and Affiliations

  1. 1.School of ScienceShandong University of TechnologyZhangdian, Zibo, ShandongP.R.China

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