Czechoslovak Mathematical Journal

, Volume 69, Issue 2, pp 337–351 | Cite as

Boundedness of Littlewood-Paley operators relative to non-isotropic dilations

  • Shuichi SatoEmail author


We consider Littlewood-Paley functions associated with a non-isotropic dilation group on ℝn. We prove that certain Littlewood-Paley functions defined by kernels with no regularity concerning smoothness are bounded on weighted Lp spaces, 1 < p < ∞, with weights of the Muckenhoupt class. This, in particular, generalizes a result of N. Rivière (1971).


Littlewood-Paley function non-isotropic dilation 

MSC 2010

42B25 46E30 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A. Benedek, A. P. Calderón, R. Panzone: Convolution operators on Banach space valued functions. Proc. Natl. Acad. Sci. USA 48 (1962), 356–365.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    A. P. Calderón: Inequalities for the maximal function relative to a metric. Stud. Math. 57 (1976), 297–306.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    A. P. Calderón, A. Torchinsky: Parabolic maximal functions associated with a distribution. Adv. Math. 16 (1975), 1–64.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    O. N. Capri: On an inequality in the theory of parabolic H p spaces. Rev. Unión Mat. Argent. 32 (1985), 17–28.MathSciNetzbMATHGoogle Scholar
  5. [5]
    L. C. Cheng: On Littlewood-Paley functions. Proc. Am. Math. Soc. 135 (2007), 3241–3247.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Y. Ding, S. Sato: Littlewood-Paley functions on homogeneous groups. Forum Math. 28 (2016), 43–55.MathSciNetzbMATHGoogle Scholar
  7. [7]
    J. Duoandikoetxea: Weighted norm inequalities for homogeneous singular integrals. Trans. Am. Math. Soc. 336 (1993), 869–880.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    J. Duoandikoetxea: Sharp L p boundedness for a class of square functions. Rev. Mat. Complut. 26 (2013), 535–548.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    J. Duoandikoetxea, J. L. Rubio de Francia: Maximal and singular integral operators via Fourier transform estimates. Invent. Math. 84 (1986), 541–561.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    J. Duoandikoetxea, E. Seijo: Weighted inequalities for rough square functions through extrapolation. Stud. Math. 149 (2002), 239–252.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    D. Fan, S. Sato: Remarks on Littlewood-Paley functions and singular integrals. J. Math. Soc. Japan 54 (2002), 565–585.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    J. Garcia-Cuerva, J. L. Rubio de Francia: Weighted Norm Inequalities and Related Topics. North-Holland Mathematics Studies 116, Notas de Matemática 104, North-Holland, Amsterdam, 1985.CrossRefzbMATHGoogle Scholar
  13. [13]
    L. Hörmander: Estimates for translation invariant operators in L p spaces. Acta Math. 104 (1960), 93–140.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    N. Riviere: Singular integrals and multiplier operators. Ark. Mat. 9 (1971), 243–278.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    J. L. Rubio de Francia: Factorization theory and A p weights. Am. J. Math. 106 (1984), 533–547.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    S. Sato: Remarks on square functions in the Littlewood-Paley theory. Bull. Aust. Math. Soc. 58 (1998), 199–211.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    S. Sato: Estimates for Littlewood-Paley functions and extrapolation. Integral Equations Oper. Theory 62 (2008), 429–440.MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    S. Sato: Estimates for singular integrals along surfaces of revolution. J. Aust. Math. Soc. 86 (2009), 413–430.MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    S. Sato: Littlewood-Paley equivalence and homogeneous Fourier multipliers. Integral Equations Oper. Theory 87 (2017), 15–44.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    E. M. Stein: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series 30, Princeton University Press, Princeton, 1970.zbMATHGoogle Scholar
  21. [21]
    E. M. Stein, S. Wainger: Problems in harmonic analysis related to curvature. Bull. Am. Math. Soc. 84 (1978), 1239–1295.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Mathematical Institute, Academy of Sciences of Cz 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of EducationKanazawa UniversityKanazawaJapan

Personalised recommendations