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# Boundedness of the $q$-Mean-Square Operator on Vector-Valued Analytic Martingales

Published:1999-06-01
Printed: Jun 1999
• Peide Liu
• Eero Saksman
• Hans-Olav Tylli
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## Abstract

We study boundedness properties of the $q$-mean-square operator $S^{(q)}$ on $E$-valued analytic martingales, where $E$ is a complex quasi-Banach space and $2 \leq q < \infty$. We establish that a.s. finiteness of $S^{(q)}$ for every bounded $E$-valued analytic martingale implies strong $(p,p)$-type estimates for $S^{(q)}$ and all $p\in (0,\infty)$. Our results yield new characterizations (in terms of analytic and stochastic properties of the function $S^{(q)}$) of the complex spaces $E$ that admit an equivalent $q$-uniformly PL-convex quasi-norm. We also obtain a vector-valued extension (and a characterization) of part of an observation due to Bourgain and Davis concerning the $L^p$-boundedness of the usual square-function on scalar-valued analytic martingales.
 MSC Classifications: 46B20 - Geometry and structure of normed linear spaces 60G46 - Martingales and classical analysis

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