A noncommutative weak type (1,1) estimate for a square function from ergodic theory

摘要

In this paper, we investigate the boundedness of a square function operator from ergodic theory acting on noncommutative L-p-spaces. The main result is a weak type (1, 1) estimate of this operator. We also show the (L-infinity, BMO) estimate, and thus all the strong type (L-p, L-p) estimates by interpolation. The main new difficulty lies in the fact that the kernel of this square function operator does not enjoy any regularity, while the Lipschitz regularity assumption is crucial in showing such endpoint estimates for the noncommutative Calderon-Zygmund singular integrals. (C) 2021 Elsevier Inc. All rights reserved.