Estimates for compositions of maximal operators with singular integrals

TL;DR

This paper establishes weak-type (1,1) bounds for compositions of maximal operators with singular integrals, improving $L^q$ bounds for certain operators and exploring variation-norm estimates.

Contribution

It provides new weak-type estimates for compositions involving Bourgain's maximal multiplier and modulated singular integrals, with improved bounds for $L^q$ norms.

Findings

Proved weak-type (1,1) estimates for the operator $ abla^* ext{ extPsi}$.

Achieved significantly improved bounds for the $L^q$ operator norm when $1<q<2$.

Explored associated variation-norm estimates.

Abstract

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