Analysis Mathematica

, Volume 43, Issue 4, pp 547–565 | Cite as

Multilinear Hardy–Cesàro operator and commutator on the product of Morrey–Herz spaces

  • N. M. Chuong
  • N. T. Hong
  • H. D. HungEmail author


We obtain necessary and sufficient conditions on functions s 1(t),..., s m (t) and ψ(t) such that the weighted multilinear Hardy–Cesàro operator
$$\left( {f_1 , \ldots f_m } \right) \mapsto \int_{[0,1]^n } {\left( {\prod\limits_{k = 1}^m {f_k } \left( {s_k \left( t \right)x} \right)} \right)\psi (t)dt}$$
is bounded from \(\dot K_{q_1 }^{\alpha _1 ,p_1 } \left( {\omega _1 } \right) \times \cdots \times \dot K_{q_m }^{\alpha _m ,p_m } \left( {\omega _m } \right)\) to \(\dot K_q^{\alpha ,p} \left( \omega \right)\) and from \(M\dot K_{p_1 ,q_1 }^{\alpha _1 ,\lambda _1 } \left( {\omega _1 } \right) \times \cdots \times M\dot K_{p_m ,q_m }^{\alpha _m ,\lambda _m } \left( {\omega _m } \right)\) to \(M\dot K_{p,q}^{\alpha ,\lambda } \left( \omega \right)\). Sharp bounds are also obtained for both cases 0 < p < 1 and 1 ≤ p < ∞. Provided b 1, ..., b m are Lipschitz functions we give a sufficient condition on functions s 1(t), ..., s m (t), ψ(t) such that the commutator of the weighted Hardy–Cesàro operator
$$\left( {f_1 , \ldots f_m } \right) \mapsto \int_{[0,1]^n } {\left( {\prod\limits_{k = 1}^m {f_k \left( {s_k \left( t \right)x} \right)} } \right)} \left( {\prod\limits_{k = 1}^m {\left( {b_k \left( x \right) - b_k \left( {s_k \left( t \right)x} \right)} \right)} } \right)\gamma (t)dt$$
is bounded from \(M\dot K_{p_1 ,q_1 }^{\alpha _1 ,\lambda _1 } \left( {\omega _1 } \right) \times \cdots \times M\dot K_{p_m ,q_m }^{\alpha _m ,\lambda _m } \left( {\omega _m } \right)\) to \(M\dot K_{p,q}^{\alpha ',\lambda } \left( \omega \right)\) for both cases 0 < p < 1 and 1 ≤ p < ∞. As a consequence, when m = n = 1 and s 1(t) = t, we obtain an improvement of a recent result by Tang, Xue and Zhou.


Hardy–Cesàro operator Morrey–Herz space 

Mathematics Subject Classification

47P05 47G10 42B35 


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  1. [1]
    N. M. Chuong, D. V. Duong and H. D. Hung, Bounds for the weighted Hardy–Cesàro operator and its commutator on Morrey–Herz type spaces, Z. Anal. Anwend., 35 (2016), 489–504.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    N. M. Chuong and H. D. Hung, Bounds of weighted Hardy–Cesàro operators on weighted Lebesgue and BMO spaces, Integral Transforms Spec. Funct., 25 (2014), 697–710.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    N. M. Chuong, H. D. Hung and N. T. Hong, Bounds of p-adic weighted Hardy–Cesàro operators and their commutators on p-adic weighted spaces of Morrey types, p-Adic Numbers Ultrametric Anal. Appl., 8 (2016), 31–44.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    R. R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. (2), 103 (1976), 611–635.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Z. W. Fu, S. L. Gong, S. Z. Lu and W. Yuan, Weighted multilinear Hardy operators and commutators, Forum Math., 27 (2015), 2825–2851.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Z. W. Fu, Z. G. Liu and S. Z. Lu, Commutators of weighted Hardy operators on Rn, Proc. Amer. Math. Soc., 137 (2009), 3319–3328.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Z. W. Fu and S. Z. Lu, A remark on weighted Hardy–Littlewood averages on Herz spaces, Advances in Math., 37 (2008), 632–636.MathSciNetGoogle Scholar
  8. [8]
    Z. W. Fu and S. Z. Lu, Commutators of generalized Hardy operators, Math. Nachr., 282 (2009), 832–845.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Z. W. Fu and S. Z. Lu, Weighted Hardy operators and commutators on Morrey spaces, Front. Math. China, 5 (2010), 531–549.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    D. Fan and F. Zhao, Product Hardy operators on Hardy spaces, Tokyo J. Math., 38 (2015), 193–209.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    D. Fan and F. Zhao, Multilinear Fractional Hausdorff operators, Acta Math. Sin., 30 (2014), 1407–1421.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    G. Gao and Y. Zhong, Some inequalities for Hausdorff operators, Math. Inequal. Appl., 17 (2014), 1061–1078.MathSciNetzbMATHGoogle Scholar
  13. [13]
    S. Gong, Z. W. Fu and B. Ma, Weighted multilinear Hardy operators on Herz-type spaces, Sci. World. J., 2014 (2014), ArticleID 420408.Google Scholar
  14. [14]
    H. D. Hung, The p-adic weighted Hardy–Cesàro operator and an application to discrete Hardy inequalities, J. Math. Anal. Appl., 409 (2014), 868–879.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    H. D. Hung and L. D. Ky, New weighted multilinear operators and commutators of Hardy–Cesàro type, Acta Math. Sci., Ser. B, Engl. Ed., 35B (2015), 1411–1425.CrossRefzbMATHGoogle Scholar
  16. [16]
    C. Kenig and E. M. Stein, Multilinear estimates and fractional integration, Math. Res. Lett., 6 (1999), 1–15.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    E. Liflyand, Hausdorff operators on Hardy spaces, Eurasian Math. J., 4 (2013), 101–141.MathSciNetzbMATHGoogle Scholar
  18. [18]
    S. Lu and L. F. Xu, Boundedness of rough singular integral operators on the homogeneous Morrey–Herz spaces, Hokkaido Math. J., 34 (2005), 299–314.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    S. Lu, D. Yang and G. Hu, Herz Type Spaces and their Applications, Science Press (Beijing, China, 2008).Google Scholar
  20. [20]
    J. Xiao, Lp and BMO bounds of weighted Hardy–Littlewood averages, J. Math. Anal. Appl., 262 (2001), 660–666.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    W. Yuan, W. Sickel and D. Yang, Morrey and Campanato Meet Besov, Lizorkin and Triebel, Lecture Notes in Math. 2005, Springer (Berlin, 2010).CrossRefzbMATHGoogle Scholar
  22. [22]
    C. Tang, F. Xue and Y. Zhou, Commutators of weighted Hardy operators on Herztype spaces, Ann. Polon. Math., 101 (2011), 267–273.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2017

Authors and Affiliations

  1. 1.Institute of MathematicsVietnamese Academy of Science and TechnologyHanoiVietnam
  2. 2.Hanoi Metropolitan UniversityHanoiVietnam
  3. 3.Applied Analysis Research Group, Faculty of Mathematics and StatisticsTon Duc Thang UniversityHo Chi Minh CityVietnam

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