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.
"Sharp inequalities for martingales with values in $\ell_\infty^N$." Electron. J. Probab. 18 1 - 19, 2013. https://doi.org/10.1214/EJP.v18-2667