On the UMD constant of the space $\ell _1^N$

Adam Osękowski¹

¹ Department of Mathematics, Informatics and Mechanics University of Warsaw Banacha 2 02-097 Warszawa, Poland

Colloquium Mathematicum, Tome 142 (2016) no. 1, pp. 135-147

Résumé

Let $N\geq 2$ be a given integer. Suppose that $df=(df_n)_{n\geq 0}$ is a martingale difference sequence with values in $\ell_1^N$ and let $(\varepsilon_n)_{n\geq 0}$ be a deterministic sequence of signs. The paper contains the proof of the estimate $$ \mathbb{P}\Bigl(\sup_{n\geq 0}\, \Bigl\|\sum_{k=0}^n \varepsilon_k df_k\Bigr\|_{\ell_1^N}\geq 1\Bigr) \leq \frac{\ln N+\ln(3\ln N)}{1-(2\ln N)^{-1}}\sup_{n\geq 0}\mathbb E \Bigl\|\sum_{k=0}^n df_k\Bigr\|_{\ell_1^N}. $$ It is shown that this result is asymptotically sharp in the sense that the least constant $C_N$ in the above estimate satisfies $\lim_{N\to \infty}C_N/\!\ln N=1$. The novelty in the proof is the explicit verification of the $\zeta$-convexity of the space $\ell_1^N$.

Détail

DOI : 10.4064/cm142-1-7

Keywords: geq given integer suppose geq martingale difference sequence values ell varepsilon geq deterministic sequence signs paper contains proof estimate mathbb bigl sup geq bigl sum varepsilon bigr ell geq bigr leq frac sup geq mathbb bigl sum bigr ell shown result asymptotically sharp sense least constant above estimate satisfies lim infty novelty proof explicit verification zeta convexity space ell