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

Burkholder, Donald

doi:10.1090/S0002-9947-01-02887-2

The best constant in the Davis inequality for the expectation of the martingale square function
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by Donald L. Burkholder PDF

Trans. Amer. Math. Soc. 354 (2002), 91-105 Request permission

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 $\sqrt 3$ 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 $[0,\infty )$. Some earlier inequalities are also improved and, closely related, the Lévy martingale is embedded in a large family of submartingales.

References

D. L. Burkholder, Martingale transforms, Ann. Math. Statist. 37 (1966), 1494–1504. MR 208647, DOI 10.1214/aoms/1177699141
Donald L. Burkholder, Sharp inequalities for martingales and stochastic integrals, Astérisque 157-158 (1988), 75–94. Colloque Paul Lévy sur les Processus Stochastiques (Palaiseau, 1987). MR 976214
Donald L. Burkholder, Sharp norm comparison of martingale maximal functions and stochastic integrals, Proceedings of the Norbert Wiener Centenary Congress, 1994 (East Lansing, MI, 1994) Proc. Sympos. Appl. Math., vol. 52, Amer. Math. Soc., Providence, RI, 1997, pp. 343–358. MR 1440921, DOI 10.1090/psapm/052/1440921
D. L. Burkholder and R. F. Gundy, Extrapolation and interpolation of quasi-linear operators on martingales, Acta Math. 124 (1970), 249–304. MR 440695, DOI 10.1007/BF02394573
Eric A. Carlen and Paul Krée, On martingale inequalities in non-commutative stochastic analysis, J. Funct. Anal. 158 (1998), no. 2, 475–508. MR 1648491, DOI 10.1006/jfan.1998.3299
David C. Cox, The best constant in Burkholder’s weak-$L^{1}$ inequality for the martingale square function, Proc. Amer. Math. Soc. 85 (1982), no. 3, 427–433. MR 656117, DOI 10.1090/S0002-9939-1982-0656117-9
Burgess Davis, On the integrability of the martingale square function, Israel J. Math. 8 (1970), 187–190. MR 268966, DOI 10.1007/BF02771313
Claude Dellacherie and Paul-André Meyer, Probabilities and potential. B, North-Holland Mathematics Studies, vol. 72, North-Holland Publishing Co., Amsterdam, 1982. Theory of martingales; Translated from the French by J. P. Wilson. MR 745449
Catherine Doléans, Variation quadratique des martingales continues à droite, Ann. Math. Statist. 40 (1969), 284–289 (French). MR 236982, DOI 10.1214/aoms/1177697823
Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
Charles Fefferman, Characterizations of bounded mean oscillation, Bull. Amer. Math. Soc. 77 (1971), 587–588. MR 280994, DOI 10.1090/S0002-9904-1971-12763-5
C. Fefferman and E. M. Stein, $H^{p}$ spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137–193. MR 447953, DOI 10.1007/BF02392215
Adriano M. Garsia, On the smoothness of functions satisfying certain integral inequalities, Functional Analysis (Proc. Sympos., Monterey, Calif., 1969) Academic Press, New York, 1969, pp. 127–162. MR 0273388
Adriano M. Garsia, Martingale inequalities: Seminar notes on recent progress, Mathematics Lecture Note Series, W. A. Benjamin, Inc., Reading, Mass.-London-Amsterdam, 1973. MR 0448538
Kiyoshi Itô and Shinzo Watanabe, Transformation of Markov processes by multiplicative functionals, Ann. Inst. Fourier (Grenoble) 15 (1965), no. fasc. 1, 13–30. MR 184282, DOI 10.5802/aif.192
A. Khintchine, Über dyadische Brüche, Math. Z. 18 (1923), 109–116.
Morgan Ward, Ring homomorphisms which are also lattice homomorphisms, Amer. J. Math. 61 (1939), 783–787. MR 10, DOI 10.2307/2371336
J. E. Littlewood, On bounded bilinear forms in an infinite number of variables, Quart. J. Math. Oxford 1 (1930), 164–174.
J. E. Littlewood, Littlewood’s miscellany, Cambridge University Press, Cambridge, 1986. Edited and with a foreword by Béla Bollobás. MR 872858
J. Marcinkiewicz, Quelques théorèmes sur les séries orthogonales, Ann. Soc. Polon. Math. 16 (1937), 84–96.
J. Marcinkiewicz and A. Zygmund, Quelques théorèmes sur les fonctions indépendantes, Studia Math. 7 (1938), 104–120.
R. E. A. C. Paley, A remarkable series of orthogonal functions I, Proc. London Math. Soc. 34 (1932), 241–264.
Gilles Pisier and Quanhua Xu, Non-commutative martingale inequalities, Comm. Math. Phys. 189 (1997), no. 3, 667–698. MR 1482934, DOI 10.1007/s002200050224
E. M. Stein, The development of square functions in the work of A. Zygmund, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 2, 359–376. MR 663787, DOI 10.1090/S0273-0979-1982-15040-6
Miloslav Duchoň, On Herglotz theorem in vector lattices, Tatra Mt. Math. Publ. 3 (1993), 231–236. Measure theory (Liptovský Ján, 1993). MR 1278537