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In this article, we prove a weak type $(p,p)$ maximal inequality,$1<p<\infty$, for weighted averages of a positive Dunford-Schwarz operator $T$acting on a noncommutative $L_p$-space associated to a semifinite von Neumannalgebra $\mathcal{M}$, with weights in $W_q$, where$\frac{1}{p}+\frac{1}{q}=1$. This result is then utilized to obtain modulatedindividual ergodic theorems with $q$-Besicovitch and $q$-Hartman sequences asweights. Multiparameter versions of these results are also investigated.

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