A new proof of certain littlewood-paley inequalities

  • Lucien Chevalier


The aim of this article is to give a new proof of the Lp-inequalities for the Littlewood-Paley g*-function. Our main tool is a pointwise equality, relating a function f, and the associated functional g*(f), which has the form f2=h(f)+g * 2 (f), where h(f) is an explicit function. We obtain this equality as a particular case of a more general one, which is reminiscent of a well-known identity in the stochastic calculus setting, namely the Itô formula. Once the above equality is proved, Lp-estimates for g*(f) are obviously equivalent to Lp/2-estimates for h(f). We obtain these last estimates (more precisely, Hp/2-estimates for h(f) by using a slight extension of the Coifman-Meyer-Stein theorem relating the so-called tent-spaces and the Hardy spaces. We observe that our methods clearly show that the restriction p>2n/n+1 is closely related to cancellation and size properties of the gradient of the Poisson kernel.

Math Subject Classifications


Key words and Phrases

Littlewood-Paley inequalities Hardy spaces 


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  1. [1]
    Chevalier, L. (1998). Une “formule de Tanaka” en analyse harmonique et quelques applications,Adv in Math.,138(1), 182–210.MATHMathSciNetCrossRefGoogle Scholar
  2. [2]
    Chevalier, L. (1999). Une renormalisation du produit,C. R. Acad. Sci. Paris,329 Série I, 265–268.MATHMathSciNetGoogle Scholar
  3. [3]
    Chevalier, L. (2000). Mouvement brownien, et formule de Tanaka en analyse,Potential Anal.,12, 419–439.MATHMathSciNetCrossRefGoogle Scholar
  4. [4]
    Coifman, R.R., Meyer, Y., and Stein, E.M. (1983). Un nouvel espace fonctionnel adapté à l'étude des opérateurs définis par des intégrales singulièresProceedings of a Conference in Harmonic Analysis held at Cortona, Lecture Notes in Math.,992, 1–15.MATHMathSciNetGoogle Scholar
  5. [5]
    Coifman, R.R., Meyer, Y., and Stein, E.M. (1985). Some new function spaces and their applications to harmonic analysis,J. Funct. Analysis,62, 304–335.MATHMathSciNetCrossRefGoogle Scholar
  6. [6]
    Fefferman, C. and Stein, E.M. (1972).H p, spaces of several variables,Acta Math.,129, 137–193.MATHMathSciNetCrossRefGoogle Scholar
  7. [7]
    Gundy, R.F. (1983).The density of the area integral, Conference on Harmonic Analysis in Honor of Antoni Zygmund. Wadsworth, Belmont, CA, 138–149.Google Scholar
  8. [8]
    Gundy, R.F. and Silverstein, M.L. (1985). The density of the area integral in ℝ+n+1,Ann. Inst. Fourier,35, 215–229.MATHMathSciNetGoogle Scholar
  9. [9]
    Meyer, Y. (1990).Ondelettes et Opérateurs I, Hermann, Paris.MATHGoogle Scholar
  10. [10]
    Stein, E.M. (1970).Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ.MATHGoogle Scholar
  11. [11]
    Stein, E.M. (1993).Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals, Princeton University Press, Princeton, NJ.Google Scholar
  12. [12]
    Zygmund, A. (1944). On certain integrals,Trans. Am. Math. Soc.,55, 170–204.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Birkhäuser Boston 2001

Authors and Affiliations

  • Lucien Chevalier
    • 1
  1. 1.Institut FourierU.M.R. 5582 C.N.R.S./U.J.F.Saint Martin d'HèresFrance

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