A note on weighted bounds for rough singular integrals

TL;DR

This paper establishes that the operator norm of the composition of a maximal operator and a rough singular integral on weighted L^2 spaces depends quadratically on the A_2 characteristic of the weight, and this dependence is proven to be optimal.

Contribution

It provides a sharp quadratic bound for the weighted norm of the composition of maximal and rough singular integral operators, extending understanding of weighted inequalities in harmonic analysis.

Findings

The operator norm depends quadratically on the A_2 characteristic.

The quadratic dependence is shown to be sharp.

The result applies to rough homogeneous singular integrals with bounded angular part.

Abstract

We show that the $L^2(w)$ operator norm of the composition $M\!\circ T_{\Omega}$, where $M$ is the maximal operator and $T_{\Omega}$ is a rough homogeneous singular integral with angular part $\Omega\in L^{\infty}(S^{n-1})$, depends quadratically on $[w]_{A_2}$, and this dependence is sharp.