Noncommutative differential transforms for averaging operators.

In this paper, we complete the study of mapping properties about the difference associated with dyadic differentiation average and dyadic martingale on noncommutative $L_{p}$-spaces. To be more precise, we establish the weak type $(1,1)$ and $(L_{\infty},\mathrm{BMO})$ estimates of this difference. Consequently, in conjunction with interpolation and duality, we obtain the corresponding all strong type $(p,p)$ estimates. As an important application, we obtain the weak type $(1,1)$ and strong type $(p,p)$ estimates of noncommutative differential transforms. The main difficulties lie in the fact that the kernel associated with considering operator does not enjoy any regularity, while the necessary regularity assumption is required to prove such endpoint estimates for the operator-valued Calder\'on-Zygmund singular integrals. Moreover, the almost orthogonality principle used in \cite{HX} is no longer applicable in the present case.