Transactions of the American Mathematical Society

The best constant in the Davis inequality for the expectation of the martingale square function

Author(s): Donald L. Burkholder
Journal: Trans. Amer. Math. Soc. 354 (2002), 91-105.
MSC (2000): Primary 60G44, 60G42; Secondary 60J65, 42B25
Posted: August 20, 2001
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Abstract: 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 $\sqrt3$ 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

indexed by $[0,\infty)$ . Some earlier inequalities are also improved and, closely related, the Lévy martingale is embedded in a large family of submartingales.

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

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 60G44, 60G42, 60J65, 42B25

Donald L. Burkholder
Affiliation: Department of Mathematics, 273 Altgeld Hall, 1409 West Green Street, University of Illinois, Urbana, Illinois 61801
Email: donburk@math.uiuc.edu

DOI: 10.1090/S0002-9947-01-02887-2
PII: S 0002-9947(01)02887-2
Keywords: Square function, maximal function, martingale
Received by editor(s): March 2, 2001
Posted: August 20, 2001
Copyright of article: Copyright 2001, American Mathematical Society

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