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

**Adam Osękowski**

*(University of Warsaw)*

#### Abstract

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.

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Pages: 1-19

Publication Date: August 9, 2013

DOI: 10.1214/EJP.v18-2667

#### References

- Burkholder, D. L. A geometrical characterization of Banach spaces in which martingale
difference sequences are unconditional.
*Ann. Probab.*9 (1981), no. 6, 997--1011. MR0632972 - Burkholder, D. L. Boundary value problems and sharp inequalities for martingale
transforms.
*Ann. Probab.*12 (1984), no. 3, 647--702. MR0744226 - 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. MR0864712 - 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. MR1108183 - Burkholder, D. L.; Gundy, R. F. Extrapolation and interpolation of quasi-linear operators on
martingales.
*Acta Math.*124 (1970), 249--304. MR0440695 - Geiss, Stefan. ${\rm BMO}_ \psi$-spaces and applications to extrapolation
theory.
*Studia Math.*122 (1997), no. 3, 235--274. MR1434474 - Lee, Jinsik Mok. On Burkholder's biconvex-function characterization of Hilbert
spaces.
*Proc. Amer. Math. Soc.*118 (1993), no. 2, 555--559. MR1159174 - Osękowski, Adam. Inequalities for dominated martingales.
*Bernoulli*13 (2007), no. 1, 54--79. MR2307394 - Osȩkowski, Adam. On relaxing the assumption of differential subordination in some
martingale inequalities.
*Electron. Commun. Probab.*16 (2011), 9--21. MR2753300

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