The best constant in the Davis inequality for the expectation of the martingale square function
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- by Donald L. Burkholder PDF
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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 $\sqrt 3$ 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
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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
- Received by editor(s): March 2, 2001
- Published electronically: August 20, 2001
- © Copyright 2001 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 354 (2002), 91-105
- MSC (2000): Primary 60G44, 60G42; Secondary 60J65, 42B25
- DOI: https://doi.org/10.1090/S0002-9947-01-02887-2
- MathSciNet review: 1859027