The Best Constant in the Davis Inequality for the Expectation of the Martingale Square Function

  • Burgess DavisEmail author
  • Renming Song
Part of the Selected Works in Probability and Statistics book series (SWPS)


A method is introduced for the simultaneous study of the square function and the maximal function of a martingale that can yield sharp norm inequalities between the two. One application is that the expectation of the square function of a martingale is not greater than \(\sqrt {3}\) times the expectation of the maximal function. This gives the best constant for one side of the Davis two-sided inequality. The martingale may take its values in any real or complex Hilbert space. The elementary discrete-time case leads quickly to the analogous results for local martingales M indexed by [0, ∞). Some earlier inequalities are also improved and, closely related, the Lévy martingale is embedded in a large family of submartingales.


Maximal Function Complex Hilbert Space Good Constant Local Martingale Pointwise Limit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    D. L. Burkholder, Martingale transforms, Ann. Math. Statist. 37(1966), 1494-1504. MR 34:8456zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    D. L. Burkholder, Sharp inequalities for martingales and stochastic integrals, Astérisque 157-158(1988), 75-94. MR 90b:60051MathSciNetGoogle Scholar
  3. 3.
    D. L. Burkholder, Sharp norm comparison of martingale maximal functions and stochastic integrals, Proceedings of the Norbert Wiener Centenary Congress, 1994 (East Lansing, MI), Proc. Sympos. Appl. Math. 52(1997), 343-358. MR 98f:60103Google Scholar
  4. 4.
    D. L. Burkholder and R. F. Gundy, Extrapolation and interpolation of quasi-linear operators on martingales, Acta Math. 124(1970), 249-304. MR 55:13567zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    E. A. Carlen and P. Krée, On martingale inequalities in non-commutative stochastic analysis, J. Funct. Anal. 158(1998), 475-508. MR 99g:81111zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    D. C. Cox, The best constant in Burkholder’s weak-L 1 inequality for the martingale square function, Proc. Amer. Math. Soc. 85(1982), 427-433. MR 84g:60079zbMATHMathSciNetGoogle Scholar
  7. 7.
    B. Davis, On the integrability of the martingale square function, Israel J. Math. 8(1970), 187-190. MR 42:3863zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    C. Dellacherie and P.-A. Meyer, Probabilities and potential B: Theory of martingales, North Holland, Amsterdam, 1982. MR 85e:60001Google Scholar
  9. 9.
    C. Doléans, Variation quadratique des martingales continues à droite, Ann. Math. Statist. 40(1969), 284-289. MR 38:5275zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    J. L. Doob, Stochastic processes, Wiley, New York, 1953. MR 15:445bGoogle Scholar
  11. 11.
    C. Fefferman, Characterization of bounded mean oscillation, Bull. Amer. Math. Soc. 77(1971), 587-588. MR 43:6713zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    C. Fefferman and E. M. Stein, H p spaces of several variables, Acta Math. 129(1972), 137-193. MR 56:6263zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    A. M. Garsia, The Burgess Davis inequalities via Fefferman’s inequality, Ark. Mat. 11(1973), 229-237. MR 42:8267zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    A. M. Garsia, Martingale Inequalities: Seminar Notes on Recent Progress, Benjamin, Reading, Massachusetts, 1973. MR 56:6844zbMATHGoogle Scholar
  15. 15.
    K. Itô and S. Watanabe, Transformation of Markov processes by multiplicative functionals, Ann. Inst. Fourier 15(1965), 15-30 MR 32:1755Google Scholar
  16. 16.
    A. Khintchine, Über dyadische Brüche, Math. Z. 18(1923), 109-116.CrossRefMathSciNetGoogle Scholar
  17. 17.
    P. Levy, Processus stochastiques et mouvement Brownien, Gauthier—Villars, Paris, 1948. MR 10:551azbMATHGoogle Scholar
  18. 18.
    J. E. Littlewood, On bounded bilinear forms in an infinite number of variables, Quart. J. Math. Oxford 1(1930), 164-174.CrossRefGoogle Scholar
  19. 19.
    J. E. Littlewood, Littlewood’s miscellany, edited and with a foreword by Béla Bollobás, Cambridge University Press, Cambridge-New York, 1986. MR 88d:01036Google Scholar
  20. 20.
    J. Marcinkiewicz, Quelques théorèmes sur les séries orthogonales, Ann. Soc. Polon. Math. 16(1937), 84-96.Google Scholar
  21. 21.
    J. Marcinkiewicz and A. Zygmund, Quelques théorèmes sur les fonctions indépendantes, Studia Math. 7(1938), 104-120.zbMATHGoogle Scholar
  22. 22.
    R. E. A. C. Paley, A remarkable series of orthogonal functions I, Proc. London Math. Soc. 34(1932), 241-264.zbMATHCrossRefGoogle Scholar
  23. 23.
    G. Pisier and Q. Xu, Non-commutative martingale inequalities, Commun. Math. Phys. 189(1997), 667-698. MR 98m:46079zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    E. M. Stein, The development of square functions in the work of A. Zygmund, Bull. Amer. Math. Soc. 7(1982), 359-376. MR 83i:42001zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993. MR 95e:42002zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Mathematics and Department of StatisticsPurdue UniversityWest LafayetteUSA
  2. 2.Department of MathematicsUniversity of IllinoisUrbanaUSA

Personalised recommendations