Sharp inequalities for martingales with values in \(\ell_\infty^N\). (English) Zbl 1287.60052

This paper provides sharp inequalities for transforms of martingales taking values in \(\ell_{\infty}^{N}\). If \(f\) is a martingale with values in \(\ell_{\infty}^{N}\) and \(g\) is its transform by a sequence of signs, then \(\|g \|_{1} \leq C_{N} \|f\|_{\infty}\). For each \(N \geq 2\), Burkholder’s method is combined with a duality argument to identify the best constant, \(C_N\), in this inequality. The result is closely related to the characterization of UMD spaces (Banach spaces satisfying the unconditional martingale differences property) in terms of \(\eta\)-convexity, studied by D. L. Burkholder [Lect. Notes Math. 1206, 61–108 (1986; Zbl 0605.60049)] and by J. M. Lee [Proc. Am. Math. Soc. 118, No. 2, 555–559 (1993; Zbl 0791.46016)].


60G42 Martingales with discrete parameter
60G44 Martingales with continuous parameter
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