  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 $(L_{\infty},\mathrm{BMO})$
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.
