  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 Moreover, the almost orthogonality principle used in \cite{HX} is no
longer applicable in the present case.
