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

   
@article{EJP2667,
	author = {Adam Osękowski},
	title = {Sharp inequalities for martingales with values in $\ell_\infty^N$},
	journal = {Electron. J. Probab.},
	fjournal = {Electronic Journal of Probability},
	volume = {18},
	year = {2013},
	keywords = {Martingale; transform; UMD space; best constants},
	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.
},
	pages = {no. 73, 1-19},
	issn = {1083-6489},
	doi = {10.1214/EJP.v18-2667},    
        url = {http://ejp.ejpecp.org/article/view/2667}}