Estimates for compositions of maximal operators with singular integrals

Abstract

We prove weak-type (1,1) estimates for compositions of maximal operators with singular integrals. Our main object of interest is the operator ${\Delta }^{\ast }\Psi$ where ${\Delta }^{\ast }$ 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.

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• Two-parameter version of Bourgain's inequality: Rational frequencies
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