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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Author: Donald L. Burkholder
Journal: Trans. Amer. Math. Soc. 354 (2002), 91-105
MSC (2000): Primary 60G44, 60G42; Secondary 60J65, 42B25
Published electronically: August 20, 2001
MathSciNet review: 1859027
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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 $M$ indexed by $[0,\infty)$. Some earlier inequalities are also improved and, closely related, the Lévy martingale is embedded in a large family of submartingales.


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Additional Information

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: http://dx.doi.org/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
Published electronically: August 20, 2001
Article copyright: © Copyright 2001 American Mathematical Society