Collectanea Mathematica

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

On generalized Littlewood–Paley functions



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.


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

Mathematics Subject Classification

Primary 42B20 Secondary 42B25 42B35 42B99 



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


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

Personalised recommendations