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

  • Liu Peide (a1), Eero Saksman (a2) and Hans-Olav Tylli (a3)

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\,\le \,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 $\text{PL}$ -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.

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References

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

  • Liu Peide (a1), Eero Saksman (a2) and Hans-Olav Tylli (a3)

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