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 $1 < q < 2$. We also consider associated variation-norm estimates.
Maximal operators on spaces BMO and BLO
Maximal operators associated with Fourier multipliers and applications
Sharp $\left( H_{p},L_{p}\right) $ type inequalities of maximal operators of $T$ means with respect to Vilenkin systems with monotone coefficients
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