Noncommutative differential transforms for averaging operators

In this paper, we complete the study of mapping properties for a family of operators evaluating the difference between differentiation operators and conditional expectations acting on noncommutative $L_p$-spaces. To be more precise, we establish the weak type $(1,1)$ and $(L_{\mathrm{\infty }},\mathrm{B}\mathrm{M}\mathrm{O})$ estimates of this difference. Consequently, in conjunction with interpolation and duality, we obtain all strong type $(p,p)$ estimates. This allows us to obtain a quick application to noncommutative differential transforms for averaging operators.