Analysis in Theory and Applications

, Volume 22, Issue 1, pp 56–64 | Cite as

Boundedness of Marcinkiewicz integral on the weighted Herz-type Hardy spaces

  • Lejin Xu


In this paper, we discuss the boundedness of Marcinkiewicz integral μΩ with homogeneous kernel on the weighted Herz-type Hardy spaces, and prove that μΩ is bounded from \(H\dot K_q^{\alpha ,p} (w_1 ;w_2 )\) into \(\dot K_q^{\alpha ,p} (w_1 ;w_2 )\).

Key words

Marcinkiewicz integral Lr-Dini condition weighted Herz-type Hardy spaces 

AMS(2000) subject classification

42B30 42B25 42B35 


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  1. [1]
    Stein, E. M., On the Function of Littlewood-Paley, Lusin and Marcinkiewicz [J], Trans. Amer. Soc., 88(1958), 430–466.CrossRefGoogle Scholar
  2. [2]
    Benedek, A., Calderon, A. P. and Panzone, R., Convolution Operators on Banach Value Functions [J], Proc. Nat. Acad. Sci. U.S.A., 48(1962), 356–365.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    Ding, Y., Fan, D.S. and Pan, Y.B.,L p-boundedness of Marcinkiewicz Integrals with Hardy Function Kernel [J], Acta Math. Sinica (English Ser.), 16(2000), 593–600.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    Torchinsky, A. and Wang, S. L., A Note on Marcinkiewicz Integral [J], Collq. Math., 61(1990), 235–243.MathSciNetGoogle Scholar
  5. [5]
    Ding, Y., Fan, D. S. and Pan, Y. B., Weighted Boundedness for a Class of Rough Marcinkiewicz Integrals [J], Indians Univ. Math. J., 48:3(1999), 1037–1055.MathSciNetzbMATHGoogle Scholar
  6. [6]
    Chen, D. X. and Zhang, P., The Marcinkiewicz Integral with Homogeneous Kernel on the Herz-type Hardy Spaces [J], Chinese Annals of Math. 25(A),3(2004), 367–372.Google Scholar
  7. [7]
    Lu, S. Z. and Yang, D. C., The Weighted Herz-type Hardy Spaces and its Applications [J], Sci. China Ser. A, 38(6)(1995), 662–673.MathSciNetzbMATHGoogle Scholar
  8. [8]
    Soria, F. and Weiss, G., A Note on Singular Integrals and Power Weights, Indiana Univ. Math. J., 43(1994), 187–204.MathSciNetzbMATHGoogle Scholar
  9. [9]
    Ding, Y. and Lu, S. Z., Homogeneous Fractional Integrals on Hardy Spaces [J], Tohoku Math. J., 52(200), 153–162.Google Scholar
  10. [10]
    Lu, S. Z. and Yang, D. C., Hardy-Littlewood-Sobolev Theorems of Fractional Integral on Herz-type Spaces and its Application [J], Canad. J. Math. 48(1996), 363–380.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  • Lejin Xu
    • 1
  1. 1.Department of MathematicsZhejiang UniversityHangzhouP.R. China

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