Abstract
Banach space $X$-valued martingale transforms by a $\mathcal{B}(X)$-valued multiplier sequence are bounded on $L_p(X)$, where $1<p<\infty$ and $X$ is a UMD space, if and only if the multiplier sequence is pointwise R-bounded. This is also true for unconditionally convergent martingales in arbitrary Banach spaces.
Citation
Maria Girardi. Lutz Weis. "Operator-valued martingale transforms and R-boundedness." Illinois J. Math. 49 (2) 487 - 516, Summer 2005. https://doi.org/10.1215/ijm/1258138030
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