Open Access -Boundedness of Marcinkiewicz Integrals along Surfaces with Variable Kernels: Another Sufficient Condition

Journal of Inequalities and Applications20072007:026765

DOI: 10.1155/2007/26765

Received: 18 December 2006

Accepted: 23 April 2007

Published: 6 June 2007


We give the estimates for the Marcinkiewicz integral with rough variable kernels associated to surfaces. More precisely, we give some other sufficient conditions which are different from the conditions known before to warrant that the -boundedness holds. As corollaries of this result, we show that similar properties still hold for parametric Littlewood-Paley area integral and parametric functions with rough variable kernels. Some of the results are extensions of some known results.


Authors’ Affiliations

School of Mathematical Sciences, Beijing Normal University
School of Science and Technology, Kwansei Gakuin University


  1. Calderón AP, Zygmund A: On a problem of Mihlin. Transactions of the American Mathematical Society 1955,78(1):209–224.MathSciNetView ArticleMATHGoogle Scholar
  2. Aguilera NE, Harboure EO: Some inequalities for maximal operators. Indiana University Mathematics Journal 1980,29(4):559–576. 10.1512/iumj.1980.29.29042MathSciNetView ArticleMATHGoogle Scholar
  3. Tang L, Yang D: Boundedness of singular integrals of variable rough Calderón-Zygmund kernels along surfaces. Integral Equations and Operator Theory 2002,43(4):488–502. 10.1007/BF01212707MathSciNetView ArticleMATHGoogle Scholar
  4. Stein EM: On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz. Transactions of the American Mathematical Society 1958,88(2):430–466. 10.1090/S0002-9947-1958-0112932-2MathSciNetView ArticleGoogle Scholar
  5. Ding Y, Fan D, Pan Y: Weighted boundedness for a class of rough Marcinkiewicz integrals. Indiana University Mathematics Journal 1999,48(3):1037–1055.MathSciNetView ArticleMATHGoogle Scholar
  6. Ding Y, Fan D, Pan Y: -boundedness of Marcinkiewicz integrals with Hardy space function kernels. Acta Mathematica Sinica 2000,16(4):593–600. 10.1007/s101140000015MathSciNetView ArticleMATHGoogle Scholar
  7. Fan D, Sato S: Weak type estimates for Marcinkiewicz integrals with rough kernels. Tohoku Mathematical Journal 2001,53(2):265–284. 10.2748/tmj/1178207481MathSciNetView ArticleMATHGoogle Scholar
  8. Ding Y, Lin C-C, Shao S: On the Marcinkiewicz integral with variable kernels. Indiana University Mathematics Journal 2004,53(3):805–821. 10.1512/iumj.2004.53.2406MathSciNetView ArticleMATHGoogle Scholar
  9. Xue Q, Yabuta K: -boundedness of Marcinkiewicz integrals along surfaces with variable kernels. Scientiae Mathematicae Japonicae 2006,63(3):369–382.MathSciNetMATHGoogle Scholar
  10. Al-Qassem HM: On weighted inequalities for parametric Marcinkiewicz integrals. Journal of Inequalities and Applications 2006, 2006: 17 pages.Google Scholar
  11. Stein EM, Weiss G: Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series, no. 32. Princeton University Press, Princeton, NJ, USA; 1971:x+297.Google Scholar
  12. Xue Q, Yabuta K: Correction and addition to " -boundedness of Marcinkiewicz integrals along surfaces with variable kernels". Scientiae Mathematicae Japonicae 2007,65(2):291–298.MathSciNetMATHGoogle Scholar
  13. Lorch L, Szego P: A singular integral whose kernel involves a Bessel function. Duke Mathematical Journal 1955,22(3):407–418. 10.1215/S0012-7094-55-02244-4MathSciNetView ArticleMATHGoogle Scholar
  14. Watson GN: A Treatise on the Theory of Bessel Functions. 2nd edition. Cambridge University Press, London, UK; 1966.MATHGoogle Scholar
  15. Calderón AP, Zygmund A: On singular integrals with variable kernels. Applicable Analysis 1978,7(3):221–238. 10.1080/00036817808839193View ArticleMATHGoogle Scholar


© Q. Xue and K. Yabuta 2007

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.