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Collectanea Mathematica

, Volume 69, Issue 2, pp 297–314 | Cite as

On generalized Littlewood–Paley functions

  • H. Al-Qassem
  • L. Cheng
  • Y. Pan
Article
  • 139 Downloads

Abstract

We study the \(L^{p}\) boundedness of certain classes of generalized Littlewood–Paley functions \(\mathcal{S}_{\Phi }^{(\lambda )}(f)\). We obtain \(L^{p}\) estimates of \(\mathcal{S}_{\Phi }^{(\lambda )}(f)\) with sharp range of p and under optimal conditions on \(\Phi \). By using these estimates along with an extrapolation argument we obtain some new and improved results on generalized Littlewood–Paley functions. The approach in proving our results is mainly based on proving vector-valued inequalities and in turn the proof of our results (in the case \(\lambda =2)\) provides us with alternative proofs of the results obtained by Duoandikoetxea as his approach is based on proving certain weighted norm inequalities.

Keywords

Littlewood–Paley functions Triebel–Lizorkin spaces Orlicz spaces Block spaces Extrapolation \(L^{p}\)boundedness 

Mathematics Subject Classification

Primary 42B20 Secondary 42B25 42B35 42B99 

Notes

Acknowledgements

The authors would like to express their gratitude to the referee for his/her very careful reading and for many important valuable comments.

References

  1. 1.
    Al-Qassem, H., Pan, Y.: On rough maximal operators and Marcinkiewicz integrals along submanifolds. Stud. Math. 190(1), 73–98 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Al-Qassem, H., Pan, Y.: On certain estimates for Marcinkiewicz integrals and extrapolation. Collect. Math. 60(2), 123–145 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Al-Qassem, H., Cheng, L.C., Pan, Y.: On rough generalized parametric Marcinkiewicz integrals. J. Math. Inequal. 11(3), 763–780 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Al-Salman, A., Al-Qassem, H., Cheng, L., Pan, Y.: \(L^{p}\) bounds for the function of Marcinkiewicz. Math. Res. Lett. 9, 697–700 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Benedek, A., Calderón, A., Panzone, R.: Convolution operators on Banach space valued functions. Proc. Nat. Acad. Sci. U.S.A. 48, 356–365 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bourgain, J.: Averages in the plane over convex curves and maximal operators. J. Anal. Math. 47, 69–85 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cheng, L.C.: On Littlewood–Paley functions. Proc. Am. Math. Soc. 135, 3241–3247 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chen, J., Fan, D., Ying, Y.: Singular integral operators on function spaces. J. Math. Anal. Appl. 276, 691–708 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Coifman, R., Weiss, G.: Extension of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83, 569–645 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ding, Y., Fan, D., Pan, Y.: On Littlewood–Paley functions and singular integrals. Hokkaido Math. J. 29, 537–552 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ding, Y., Sato, S.: Littlewood–Paley functions on homogeneous groups. Forum Math. 28, 43–55 (2014)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Duoandikoetxea, J.: Sharp \(L^{p}\) boundedness for a class of square functions. Rev. Mat. Complut. 26(2), 535–548 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Duoandikoetxea, J., Rubio de Francia, J.L.: Maximal functions and singular integral operators via Fourier transform estimates. Invent. Math. 84, 541–561 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Fan, D., Sato, S.: Remarks on Littlewood–Paley functions and singular integrals. J. Math. Soc. Jpn. 54(3), 565–585 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Fan, D., Wu, H.: On the generalized Marcinkiewicz integral operators with rough kernels. Can. Math. Bull. 54(1), 100–112 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Iwaniec, T., Onninen, J.: \(H^{1}\)-estimates of Jacobians by subdeterminants. Math. Ann. 324(2), 341–358 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Keitoku, M., Sato, E.: Block spaces on the unit sphere in \({ R} ^{n}\). Proc. Am. Math. Soc. 119, 453–455 (1993)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Le, H.V.: Singular integrals with mixed homogeneity in Triebel–Lizorkin spaces. J. Math. Anal. Appl. 345, 903–916 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lu, S., Taibleson, M., Weiss, G.: Spaces Generated by Blocks. Beijing Normal University Press, Beijing (1989)zbMATHGoogle Scholar
  20. 20.
    Sato, S.: Remarks on square functions in the Littlewood–Paley theory. Bull. Aust. Math. Soc. 58, 199–211 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Sato, S.: Estimates for Littlewood–Paley functions and extrapolation. Integral Equ. Oper. Theory 62(3), 429–440 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Sawano, Y., Yabuta, K.: Fractional type Marcinkiewicz integral operators associated to surfaces. J. Inequal. Appl. 2014, 232 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Stein, E.M.: On the functions of Littlewood–Paley, Lusin and Marcinkiewicz. Trans. Am. Math. Soc. 88, 430–466 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Stein, E.M.: The development of square functions in the work of Zygmund. Bull. Am. Math. Soc. 7, 359–376 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Stein, E.M.: Maximal functions. I. Spherical means. Proc. Nat. Acad. Sci. U.S.A 73(7), 2174–2175 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Yabuta, K.: Triebel–Lizorkin space boundedness of Marcinkiewicz integrals associated to surfaces. Appl. Math. A J. Chin. Univ. 30(4), 418–446 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Yano, S.: An extrapolation theorem. J. Math. Soc. Jpn. 3, 296–305 (1951)CrossRefzbMATHGoogle Scholar
  28. 28.
    Walsh, T.: On the function of Marcinkiewicz. Stud. Math. 44, 203–217 (1972)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Universitat de Barcelona 2017

Authors and Affiliations

  1. 1.Department of Mathematics and PhysicsQatar UniversityDohaQatar
  2. 2.Department of MathematicsBryn Mawr CollegeBryn MawrUSA
  3. 3.Department of MathematicsUniversity of PittsburghPittsburghUSA

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