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2013 Sharp inequalities for martingales with values in $\ell_\infty^N$
Adam Osękowski
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Electron. J. Probab. 18: 1-19 (2013). DOI: 10.1214/EJP.v18-2667


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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Adam Osękowski. "Sharp inequalities for martingales with values in $\ell_\infty^N$." Electron. J. Probab. 18 1 - 19, 2013.


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

zbMATH: 1287.60052
MathSciNet: MR3091719
Digital Object Identifier: 10.1214/EJP.v18-2667

Primary: 60G42
Secondary: 60G44

Keywords: best constants , martingale , transform , UMD space

Vol.18 • 2013
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