Abstract

Bennett, DeVore and Sharpley (Ann of Math. 113: 601-611, 1981) introduced the weak analogueof the space Lo and studied its relationship to the space of functions of bounded mean oscillation.The purpose of this paper is to continue this line of research in the context of functions on Rd withvalues in a semifinite von Neumann algebra. As a by-product, this allows for the comparison of theBMO norms of an operator-valued function and its decreasing rearrangement. The argumentrests on a new distributional estimate for noncommutative martingales invoking Cuculescuprojections, which is of independent interest. The applications include related BMOwL∞inequalities for square functions and conditional square functions, as well as correspondingversions of Stein and dual Doob estimates, which are new even for classical martingales.