Boundedness of the q-Mean-Square Operator on Vector-Valued Analytic Martingales
Canadian mathematical bulletin, Tome 42 (1999) no. 2, pp. 221-230
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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.
Peide, Liu; Saksman, Eero; Tylli, Hans-Olav. Boundedness of the q-Mean-Square Operator on Vector-Valued Analytic Martingales. Canadian mathematical bulletin, Tome 42 (1999) no. 2, pp. 221-230. doi: 10.4153/CMB-1999-027-3
@article{10_4153_CMB_1999_027_3,
author = {Peide, Liu and Saksman, Eero and Tylli, Hans-Olav},
title = {Boundedness of the {q-Mean-Square} {Operator} on {Vector-Valued} {Analytic} {Martingales}},
journal = {Canadian mathematical bulletin},
pages = {221--230},
year = {1999},
volume = {42},
number = {2},
doi = {10.4153/CMB-1999-027-3},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1999-027-3/}
}
TY - JOUR AU - Peide, Liu AU - Saksman, Eero AU - Tylli, Hans-Olav TI - Boundedness of the q-Mean-Square Operator on Vector-Valued Analytic Martingales JO - Canadian mathematical bulletin PY - 1999 SP - 221 EP - 230 VL - 42 IS - 2 UR - http://geodesic.mathdoc.fr/articles/10.4153/CMB-1999-027-3/ DO - 10.4153/CMB-1999-027-3 ID - 10_4153_CMB_1999_027_3 ER -
%0 Journal Article %A Peide, Liu %A Saksman, Eero %A Tylli, Hans-Olav %T Boundedness of the q-Mean-Square Operator on Vector-Valued Analytic Martingales %J Canadian mathematical bulletin %D 1999 %P 221-230 %V 42 %N 2 %U http://geodesic.mathdoc.fr/articles/10.4153/CMB-1999-027-3/ %R 10.4153/CMB-1999-027-3 %F 10_4153_CMB_1999_027_3
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