We prove weak-type $(1,1)$ estimates for compositions of maximal
operators with singular integrals. Our main object of interest is the
operator $\Delta^*\Psi$ where $\Delta^*$ is Bourgain's maximal
multiplier operator and $\Psi$ is the sum of several modulated
singular integrals; here our method yields a significantly improved
bound for the $L^q$ operator norm when $1 \lt q \lt 2.$ We also consider
associated variation-norm estimates.