Advertisement

Lp Estimates of certain rough parametric Marcinkiewicz functions

  • Laith Hawawsheh
  • Ahmad Al-SalmanEmail author
Research Article
  • 2 Downloads

Abstract

We prove Lp estimates of a class of parametric Marcinkiewicz integral operators when their kernels satisfy only the \(L^1(\mathbb{S}^{n-1})\) integrability condition. The obtained Lp estimates resolve a problem left open in previous work. Our argument is based on duality technique and direct estimation of operators. As a consequence of our result, we deduce the Lp boundedness of a class of fractional Marcinkiewicz integral operators.

Keywords

Marcinkiewicz integrals parametric Marcinkiewicz functions rough kernels Marcinkiewicz interpolation theorem 

MSC

42B20 42B15 42B25 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The corresponding author acknowledges the support of Sultan Qaboos University through the grant IG/SCI/DOMS/18/01.

References

  1. 1.
    Al-Salman A. Marcinkiewicz functions along flat surfaces with Hardy space kernels. J Integral Equations Appl, 2005, 17(4): 357–373MathSciNetCrossRefGoogle Scholar
  2. 2.
    Al-Salman A. A class of Marcinkiewicz type integral operators. Commun Math Anal, 2012, 13(2): 56–81MathSciNetzbMATHGoogle Scholar
  3. 3.
    Al-Salman A, Al-Qassem H. On the L p boundedness of rough parametric Marcinkiewicz functions. J Inequal Pure and Appl Math, 2007, 8(4): Art 108, 8ppzbMATHGoogle Scholar
  4. 4.
    Al-Salman A, Al-Qassem H, Cheng L, Pan Y. L p bounds for the function of Marcinkiewicz. Math Res Lett, 2002, 9: 697–700MathSciNetCrossRefGoogle Scholar
  5. 5.
    Benedek A, Calderón A, Panzone R. Convolution operators on Banach space valued functions. Proc Nat Acad Sci USA, 1962, 48: 356–365MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ding Y, Lu S, Yabuta K. A problem on rough parametric Marcinkiewicz functions. J Aust Math Soc, 2002, 72: 13–21MathSciNetCrossRefGoogle Scholar
  7. 7.
    Hawawsheh L, Al-Salman A, Momani S. A note on rough parametric Marcinkiewicz functions. Anal Theory Appl (to appear)Google Scholar
  8. 8.
    Hörmander. Translation invariant operators. Acta Math, 1960, 104: 93–139MathSciNetCrossRefGoogle Scholar
  9. 9.
    Marcinkiewicz J. Sur quelques integrals de type de Dini. Ann Soc Polon Math, 1938, 17: 42–50zbMATHGoogle Scholar
  10. 10.
    Stein E M. On the functions of Littlewood-Paley, Lusin and Marcinkiewicz. Trans Amer Math Soc, 1958, 88: 430–466MathSciNetCrossRefGoogle Scholar
  11. 11.
    Stein E M. Singular Integrals and Differentiability Properties of Functions. Princeton: Princeton Univ Press, 1970zbMATHGoogle Scholar
  12. 12.
    Stein E M. Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals. Princeton: Princeton Univ Press, 1993zbMATHGoogle Scholar
  13. 13.
    Walsh T. On the function of Marcinkiewicz. Studia Math, 1972, 44: 203–217MathSciNetCrossRefGoogle Scholar

Copyright information

© Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Basic Sciences and HumanitiesGerman Jordanian UniversityAmmanJordan
  2. 2.Department of MathematicsSultan Qaboos UniversityMuscatOman

Personalised recommendations