  The purpose of the paper is to establish weighted maximal $L_p$-inequalities
in the context of operator-valued martingales on semifinite von Neumann
algebras. The main emphasis is put on the optimal dependence of the $L_p$
constants on the characteristic of the weight involved. As applications, we
establish weighted estimates for the noncommutative version of Hardy-Littlewood
maximal operator and weighted bounds for noncommutative maximal truncations of
a wide class of singular integrals.
