Séminaire de Probabilités XLVI pp 473-479

Part of the Lecture Notes in Mathematics book series (LNM, volume 2123) | Cite as

A Short Proof of Stein’s Universal Multiplier Theorem



We give a short proof of Stein’s universal multiplier theorem, purely by probabilistic methods, thus avoiding any use of harmonic analysis techniques (complex interpolation or transference methods).


  1. 1.
    M.G. Cowling, On Littlewood-Paley-Stein Theory. Proceedings of the Seminar on Harmonic Analysis (Pisa, 1980), 1981Google Scholar
  2. 2.
    R.R. Coifman, R. Rochberg, G. Weiss, Applications of Transference: The L p Version of von Neumann’s Inequality and the Littlewood-Paley-Stein Theory. Linear Spaces and Approximation (Birkhäuser, Basel, 1978)MATHGoogle Scholar
  3. 3.
    R.R. Coifman, G. Weiss, Transference Methods in Analysis (American Mathematical Society, Providence, 1976)Google Scholar
  4. 4.
    G.E. Karadzhov, M. Milman, Extrapolation theory: new results and applications. J. Approx. Theory 133(1), 38–99 (2005)CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    E. Lenglart, D. Lépingle, M. Pratelli, Présentation unifiée de certaines inégalités de la théorie des martingales. Séminaire de Probabilités, XIV, Lecture Notes in Math., vol. 784 (Springer, Berlin, 1980)Google Scholar
  6. 6.
    S. Meda, On the Littlewood-Paley-Stein g-function. Trans. Am. Math. Soc. 347(6), 2201–2212 (1995)MathSciNetMATHGoogle Scholar
  7. 7.
    P.A. Meyer, Démonstration probabiliste de certaines inégalités de Littlewood-Paley. I. Les inégalités classiques. Séminaire de Probabilités, X, Lecture Notes in Math., vol. 511 (Springer, Berlin, 1976), pp. 125–141Google Scholar
  8. 8.
    P.-A. Meyer, Sur la théorie de Littlewood-Paley-Stein (d’après Coifman-Rochberg-Weiss et Cowling). Séminaire de probabilités, XIX, Lecture Notes in Math., vol. 1123 (Springer, Berlin, 1985), pp. 113–129Google Scholar
  9. 9.
    I. Shigekawa, The Meyer Inequality for the Ornstein-Uhlenbeck Operator in L 1 and Probabilistic Proof of Stein’s L p Multiplier Theorem. Trends in Probability and Related Analysis (Taipei, 1996) (World Scientific, River Edge, 1997), pp. 273–288Google Scholar
  10. 10.
    E.M. Stein, Topics in Harmonic Analysis Related to the Littlewood-Paley Theory. Annals of Mathematics Studies, vol. 63 (Princeton University Press, Princeton, 1970)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Scuola Normale SuperiorePisaItaly

Personalised recommendations