Estimates for compositions of maximal operators with singular integrals

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