A noncommutative weak type maximal inequality for modulated ergodic averages with general weights

We prove a weak type $(p,p)$ maximal inequality, $1 \lt p \lt \infty $, for weighted averages of a positive Dunford–Schwartz operator $T$ acting on a noncommutative $L_p$-space associated to a semifinite von Neumann algebra $\mathcal M$, with weights in $W_q$, where $1/p+1/q=1$. This result is then utilized to obtain modulated individual ergodic theorems with $q$-Besicovitch and $q$-Hartman sequences as weights. Multiparameter versions of these results are also investigated.