Skip to main content
  • Research Article
  • Open access
  • Published:

A unifying approach for certain class of maximal functions

Abstract

We establish estimates for certain class of maximal functions with kernels in. As a consequence of such estimates, we obtain the boundedness of our maximal functions when their kernels are in or in the block space,. Several applications of our results are also presented.

[123456789101112131415161718192021222324252627]

References

  1. Al-Salman A: Rough oscillatory singular integral operators of nonconvolution type. Journal of Mathematical Analysis and Applications 2004,299(1):72–88. 10.1016/j.jmaa.2004.06.006

    Article  MathSciNet  MATH  Google Scholar 

  2. Al-Salman A: On maximal functions with rough kernels in. Collectanea Mathematica 2005,56(1):47–56.

    MathSciNet  MATH  Google Scholar 

  3. Al-Salman A: On a class of singular integral operators with rough kernels. Canadian Mathematical Bulletin 2006,49(1):3–10. 10.4153/CMB-2006-001-9

    Article  MathSciNet  MATH  Google Scholar 

  4. Al-Salman A, Al-Jarrah A: Rough oscillatory singular integral operators. II. Turkish Journal of Mathematics 2003,27(4):565–579.

    MathSciNet  MATH  Google Scholar 

  5. Al-Salman A, Pan Y: Singular integrals with rough kernels in. Journal of the London Mathematical Society. Second Series 2002,66(1):153–174. 10.1112/S0024610702003241

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen J, Fan DS, 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;2-0

    Article  MathSciNet  MATH  Google Scholar 

  7. Chen L-K, Lin H: A maximal operator related to a class of singular integrals. Illinois Journal of Mathematics 1990,34(1):120–126.

    MathSciNet  MATH  Google Scholar 

  8. Ding Y, Lu S, Yabuta K: A problem on rough parametric Marcinkiewicz functions. Journal of the Australian Mathematical Society 2002,72(1):13–21. 10.1017/S1446788700003542

    Article  MathSciNet  MATH  Google Scholar 

  9. Duoandikoetxea J: Weighted norm inequalities for homogeneous singular integrals. Transactions of the American Mathematical Society 1993,336(2):869–880. 10.2307/2154381

    Article  MathSciNet  MATH  Google Scholar 

  10. Fan DS, Guo K, Pan Y: estimates for singular integrals associated to homogeneous surfaces. Journal für die reine und angewandte Mathematik 2002, 542: 1–22.

    Article  MathSciNet  MATH  Google Scholar 

  11. Fan DS, Pan Y: Boundedness of certain oscillatory singular integrals. Studia Mathematica 1995,114(2):105–116.

    MathSciNet  MATH  Google Scholar 

  12. Fan DS, Pan Y: Singular integral operators with rough kernels supported by subvarieties. American Journal of Mathematics 1997,119(4):799–839. 10.1353/ajm.1997.0024

    Article  MathSciNet  MATH  Google Scholar 

  13. Fan DS, Pan Y, Yang D: A weighted norm inequality for rough singular integrals. The Tohoku Mathematical Journal. Second Series 1999,51(2):141–161.

    Article  MathSciNet  MATH  Google Scholar 

  14. 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 

  15. Hörmander L: Estimates for translation invariant operators inspaces. Acta Mathematica 1960, 104: 93–140. 10.1007/BF02547187

    Article  MathSciNet  MATH  Google Scholar 

  16. Jiang YS, Lu SZ: Oscillatory singular integrals with rough kernel. In Harmonic Analysis in China, Math. Appl.. Volume 327. Edited by: Cheng MD, Deng DG, Gong S, Yang C-C. Kluwer Academic, Dordrecht; 1995:135–145.

    Chapter  Google Scholar 

  17. Keitoku M, Sato E: Block spaces on the unit sphere in. Proceedings of the American Mathematical Society 1993,119(2):453–455.

    MathSciNet  MATH  Google Scholar 

  18. Lu S, Taibleson M, Weiss G: Spaces Generated by Blocks. Beijing Normal University Press, Beijing; 1989.

    Google Scholar 

  19. Lu SZ, Zhang Y: Criterion on-boundedness for a class of oscillatory singular integrals with rough kernels. Revista Matemática Iberoamericana 1992,8(2):201–219.

    MATH  Google Scholar 

  20. Mizuhara T: Boundedness of some classical operators on generalized Morrey spaces. In Harmonic Analysis (Sendai, 1990), ICM-90 Satell. Conference Proceedings. Edited by: Igari S. Springer, Tokyo; 1991:183–189.

    Chapter  Google Scholar 

  21. Morrey CB Jr.: On the solutions of quasi-linear elliptic partial differential equations. Transactions of the American Mathematical Society 1938,43(1):126–166. 10.1090/S0002-9947-1938-1501936-8

    Article  MathSciNet  Google Scholar 

  22. Pan Y: estimates for convolution operators with oscillating kernels. Mathematical Proceedings of the Cambridge Philosophical Society 1993,113(1):179–193. 10.1017/S0305004100075873

    Article  MathSciNet  MATH  Google Scholar 

  23. Ricci F, Stein EM: Harmonic analysis on nilpotent groups and singular integrals. I. Oscillatory integrals. Journal of Functional Analysis 1987,73(1):179–194. 10.1016/0022-1236(87)90064-4

    Article  MathSciNet  MATH  Google Scholar 

  24. Sjölin P: Convolution with oscillating kernels. Indiana University Mathematics Journal 1981,30(1):47–55. 10.1512/iumj.1981.30.30004

    Article  MathSciNet  MATH  Google Scholar 

  25. Stein EM: On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz. Transactions of the American Mathematical Society 1958,88(2):430–466. 10.1090/S0002-9947-1958-0112932-2

    Article  MathSciNet  Google Scholar 

  26. Stein EM: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series. Volume 43. Princeton University Press, New Jersey; 1993:xiv+695.

    Google Scholar 

  27. Yano S: Notes on Fourier analysis. XXIX. An extrapolation theorem. Journal of the Mathematical Society of Japan 1951, 3: 296–305. 10.2969/jmsj/00320296

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ahmad Al-Salman.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

About this article

Cite this article

Al-Salman, A. A unifying approach for certain class of maximal functions. J Inequal Appl 2006, 56272 (2006). https://doi.org/10.1155/JIA/2006/56272

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1155/JIA/2006/56272

Keywords