The composition of singular integral operators with nonsmooth kernels

TL;DR

This paper investigates the composition of two nonsmooth kernel singular integral operators, establishing quantitative bounds and weighted endpoint estimates using sparse domination techniques.

Contribution

It introduces bi-sublinear sparse domination to analyze the composition of nonsmooth kernel operators, providing new weighted bounds and endpoint estimates.

Findings

Quantitative bounds on $L^p(w)$ for the composite operator

Weighted weak type endpoint estimates

Application of sparse domination techniques

Abstract

Let $T_1$, $T_2$ be two singular integral operators with nonsmooth kernels introduced by Duong and McIntosh. In this paper, by establishing certain bi-sublinear sparse domination, the authors obtain some quantitative bounds on $L^p(\mathbb{R}^n,\,w)$ with $p\in(1,\,\infty)$ and $w\in A_p(\mathbb{R}^n)$ for the composite operator $T_1T_2$. Some weighted weak type endpoint estimates are also given.