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

Canadian mathematical bulletin

关键词: Ricci curvature;    conjugate radius;   

DOI  :  10.4153/CMB-1999-027-3

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

We study boundedness properties of the $q$-mean-square operator$S^{(q)}$ on $E$-valued analytic martingales, where $E$ is acomplex quasi-Banach space and $2 leq q < infty$. We establishthat a.s. finiteness of $S^{(q)}$ for every bounded $E$-valuedanalytic martingale implies strong $(p,p)$-type estimates for$S^{(q)}$ and all $pin (0,infty)$. Our results yield newcharacterizations (in terms of analytic and stochastic propertiesof the function $S^{(q)}$) of the complex spaces $E$ that admit anequivalent $q$-uniformly PL-convex quasi-norm. We also obtain avector-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-valuedanalytic martingales.

【 授权许可】

Unknown