Open Access

On weighted inequalities for parametric Marcinkiewicz integrals

Journal of Inequalities and Applications20062006:91541

DOI: 10.1155/JIA/2006/91541

Received: 25 February 2005

Accepted: 3 July 2005

Published: 20 March 2006

Abstract

We establish a weighted https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F91541/MediaObjects/13660_2005_Article_1656_IEq1_HTML.gif boundedness of a parametric Marcinkiewicz integral operator https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F91541/MediaObjects/13660_2005_Article_1656_IEq2_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F91541/MediaObjects/13660_2005_Article_1656_IEq3_HTML.gif is allowed to be in the block space https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F91541/MediaObjects/13660_2005_Article_1656_IEq4_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F91541/MediaObjects/13660_2005_Article_1656_IEq5_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F91541/MediaObjects/13660_2005_Article_1656_IEq6_HTML.gif satisfies a mild integrability condition. We apply this conclusion to obtain the weighted https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F91541/MediaObjects/13660_2005_Article_1656_IEq7_HTML.gif boundedness for a class of the parametric Marcinkiewicz integral operators https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F91541/MediaObjects/13660_2005_Article_1656_IEq8_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F91541/MediaObjects/13660_2005_Article_1656_IEq9_HTML.gif related to the Littlewood-Paley https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F91541/MediaObjects/13660_2005_Article_1656_IEq10_HTML.gif -function and the area integral https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F91541/MediaObjects/13660_2005_Article_1656_IEq11_HTML.gif , respectively. It is known that the condition https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F91541/MediaObjects/13660_2005_Article_1656_IEq12_HTML.gif is optimal for the https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F91541/MediaObjects/13660_2005_Article_1656_IEq13_HTML.gif boundedness of https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F91541/MediaObjects/13660_2005_Article_1656_IEq14_HTML.gif .

[12345678910111213141516171819]

Authors’ Affiliations

(1)
Department of Mathematics, Yarmouk University

References

  1. Al-Qassem HM: https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F91541/MediaObjects/13660_2005_Article_1656_IEq15_HTML.gif estimates for rough parametric Marcinkiewicz integrals. SUT Journal of Mathematics 2004,40(2):117–131.MathSciNetMATHGoogle Scholar
  2. Al-Qassem HM, Al-Salman AJ: A note on Marcinkiewicz integral operators. Journal of Mathematical Analysis and Applications 2003,282(2):698–710. 10.1016/S0022-247X(03)00244-0MathSciNetView ArticleMATHGoogle Scholar
  3. Benedek A, Calderón A-P, Panzone R: Convolution operators on Banach space valued functions. Proceedings of the National Academy of Sciences of the United States of America 1962, 48: 356–365. 10.1073/pnas.48.3.356MathSciNetView ArticleMATHGoogle Scholar
  4. Chen J, Fan D, Pan Y: A note on a Marcinkiewicz integral operator. Mathematische Nachrichten 2001,227(1):33–42. 10.1002/1522-2616(200107)227:1<33::AID-MANA33>3.0.CO;2-0MathSciNetView ArticleMATHGoogle 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. Duoandikoetxea J: Weighted norm inequalities for homogeneous singular integrals. Transactions of the American Mathematical Society 1993,336(2):869–880. 10.2307/2154381MathSciNetView ArticleMATHGoogle Scholar
  7. Duoandikoetxea J, Rubio de Francia JL: Maximal and singular integral operators via Fourier transform estimates. Inventiones Mathematicae 1986,84(3):541–561. 10.1007/BF01388746MathSciNetView ArticleMATHGoogle Scholar
  8. Fan D, Pan Y, Yang D: A weighted norm inequality for rough singular integrals. The Tohoku Mathematical Journal. Second Series 1999,51(2):141–161.MathSciNetView ArticleMATHGoogle Scholar
  9. García-Cuerva J, Rubio de Francia JL: Weighted Norm Inequalities and Related Topics, North-Holland Mathematics Studies. Volume 116. North-Holland, Amsterdam; 1985:x+604.Google Scholar
  10. Hörmander L: Estimates for translation invariant operators inspaces https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F91541/MediaObjects/13660_2005_Article_1656_IEq16_HTML.gif . Acta Mathematica 1960, 104: 93–140. 10.1007/BF02547187MathSciNetView ArticleMATHGoogle Scholar
  11. Keitoku M, Sato E: Block spaces on the unit sphere in https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F91541/MediaObjects/13660_2005_Article_1656_IEq17_HTML.gif . Proceedings of the American Mathematical Society 1993,119(2):453–455.MathSciNetMATHGoogle Scholar
  12. Kurtz DS: Littlewood-Paley and multiplier theorems on weighted https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F91541/MediaObjects/13660_2005_Article_1656_IEq18_HTML.gif spaces. Transactions of the American Mathematical Society 1980,259(1):235–254.MathSciNetMATHGoogle Scholar
  13. Lee M-Y, Lin C-C: Weighted https://static-content.springer.com/image/art%3A10.1155%2FJIA%2F2006%2F91541/MediaObjects/13660_2005_Article_1656_IEq19_HTML.gif boundedness of Marcinkiewicz integral. Integral Equations and Operator Theory 2004,49(2):211–220. 10.1007/s00020-002-1204-xMathSciNetView ArticleMATHGoogle Scholar
  14. Lu S, Taibleson M, Weiss G: Spaces Generated by Blocks. Normal University Press, Beijing; 1989.MATHGoogle Scholar
  15. Sakamoto M, Yabuta K: Boundedness of Marcinkiewicz functions. Studia Mathematica 1999,135(2):103–142.MathSciNetMATHGoogle Scholar
  16. Sato S: Remarks on square functions in the Littlewood-Paley theory. Bulletin of the Australian Mathematical Society 1998,58(2):199–211. 10.1017/S0004972700032172MathSciNetView ArticleMATHGoogle Scholar
  17. Stein EM: On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz. Transactions of the American Mathematical Society 1958, 88: 430–466. 10.1090/S0002-9947-1958-0112932-2MathSciNetView ArticleMATHGoogle Scholar
  18. Stein EM: Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, no. 30. Princeton University Press, New Jersey; 1970:xiv+290.Google Scholar
  19. Torchinsky A, Wang SL: A note on the Marcinkiewicz integral. Colloquium Mathematicum 1990,60/61(1):235–243.MathSciNetMATHGoogle Scholar

Copyright

© H.M. Al-Qassem. 2006

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.