Noncommutative weak $(1,1)$ type estimate for a square function from ergodic theory

Abstract

In this paper, we investigate the boundedness of a square function from ergodic theory on noncommutative $L_p$-spaces. The main result is a weak $(1,1)$ type estimate of this square function. We also show the $(L_{\infty },\mathrm{BMO})$ estimate, and thus strong $(L_p,L_p)$ estimate by interpolation. The main novel difficulty lies in the fact that the kernel of this square function does not enjoy any regularity, which is crucial in showing such endpoint estimates for standard noncommutative Calderón-Zygmund singular integral operators.

Based the feedbacks from colleagues, we improved substantially the presentation of the whole paper, especially the lengthy proof on weak type (1,1) estimate; among many others, in particular, we add Remark 3.8 on the geometric argument which help a lot in clarifying the rest of the proof