Electronic Journal of Probability

Sharp inequalities for martingales with values in $\ell_\infty^N$

Adam Osękowski

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The objective of the paper is to study sharp inequalities for transforms of martingales taking values in $\ell_\infty^N$. Using Burkholder's method combined with an intrinsic duality argument, we identify, for each $N\geq 2$, the best constant $C_N$ such that the following holds. If $f$ is a martingale with values in $\ell_\infty^N$ and $g$ is its transform by a sequence of signs, then

$$||g||_1\leq C_N ||f||_\infty.$$

This is closely related to the characterization of UMD spaces in terms of the so-called $\eta$ convexity, studied in the eighties by Burkholder and Lee.

Article information

Electron. J. Probab., Volume 18 (2013), paper no. 73, 19 pp.

Accepted: 9 August 2013
First available in Project Euclid: 4 June 2016

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G42: Martingales with discrete parameter
Secondary: 60G44: Martingales with continuous parameter

Martingale transform UMD space best constants

This work is licensed under a Creative Commons Attribution 3.0 License.


Osękowski, Adam. Sharp inequalities for martingales with values in $\ell_\infty^N$. Electron. J. Probab. 18 (2013), paper no. 73, 19 pp. doi:10.1214/EJP.v18-2667. https://projecteuclid.org/euclid.ejp/1465064298

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  • Burkholder, D. L. A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional. Ann. Probab. 9 (1981), no. 6, 997–1011.
  • Burkholder, D. L. Boundary value problems and sharp inequalities for martingale transforms. Ann. Probab. 12 (1984), no. 3, 647–702.
  • Burkholder, Donald L. Martingales and Fourier analysis in Banach spaces. Probability and analysis (Varenna, 1985), 61–108, Lecture Notes in Math., 1206, Springer, Berlin, 1986.
  • Burkholder, Donald L. Explorations in martingale theory and its applications. École d'Été de Probabilités de Saint-Flour XIX–1989, 1–66, Lecture Notes in Math., 1464, Springer, Berlin, 1991.
  • Burkholder, D. L.; Gundy, R. F. Extrapolation and interpolation of quasi-linear operators on martingales. Acta Math. 124 (1970), 249–304.
  • Geiss, Stefan. ${\rm BMO}_ \psi$-spaces and applications to extrapolation theory. Studia Math. 122 (1997), no. 3, 235–274.
  • Lee, Jinsik Mok. On Burkholder's biconvex-function characterization of Hilbert spaces. Proc. Amer. Math. Soc. 118 (1993), no. 2, 555–559.
  • Osȩkowski, Adam. Inequalities for dominated martingales. Bernoulli 13 (2007), no. 1, 54–79.
  • Osȩkowski, Adam. On relaxing the assumption of differential subordination in some martingale inequalities. Electron. Commun. Probab. 16 (2011), 9–21.