Noncommutative weak-$L^\infty$ and BMO
Organizers
Speaker
Lian Wu
Time
Wednesday, March 13, 2024 9:00 AM - 12:00 PM
Venue
A3-3-301
Online
Zoom 242 742 6089
(BIMSA)
Abstract
Bennett, DeVore and Sharpley (Ann of 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 $R^d$ with values in a semifinite 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 new distributional estimate for noncommutative martingales invoking Cuculescu projections, which is of independent interest. The applications include related $BMO\to wL^\infty$ 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.