Bennett et al. (Ann Math 113:601–611, 1981) introduced the weak analogue of the space \(L^\infty \) 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 \(\mathbb {R}^d\) with values in a finite von Neumann algebra. As a by-product, this allows for the comparison of the BMO norms of an operator-valued function and its decreasing rearrangement. The argument rests on a novel distributional estimate for noncommutative martingales invoking Cuculescu projections, which is of independent interest. The applications include related \(BMO\rightarrow wL^\infty \) inequalities for square functions and conditional square functions, as well as corresponding versions of Stein and dual Doob estimates, which are new even for classical martingales.
