Quantitative weighted estimates for rough singular integrals on homogeneous groups

TL;DR

This paper establishes quantitative weighted bounds for rough singular integrals on homogeneous groups, extending Euclidean results to more general spaces with weaker assumptions, and also addresses bi-parameter cases on product groups.

Contribution

It provides the first quantitative weighted estimates for rough singular integrals on homogeneous groups, generalizing Euclidean results and considering bi-parameter operators.

Findings

Weighted bounds match Euclidean case results

Weaker assumptions on kernels and spaces compared to prior work

Quantitative bounds for bi-parameter rough singular integrals

Abstract

In this paper, we study weighted $L^{p}(w)$ boundedness ($1<p<\infty$ and $w$ a Muckenhoupt $A_{p}$ weight) of singular integrals with homogeneous convolution kernel $K(x)$ on an arbitrary homogeneous group $\mathbb H$ of dimension $\mathbb{Q}$, {under the assumption that $K_0$, the restriction of $K$ to the unit annulus, is mean zero and $L^{q}$ integrable for some $q_{0}<q\leq \infty$,} where $q_{0}$ is a fixed constant depending on $w$. We obtain a quantitative weighted bound, which is consistent with the one obtained by Hyt\"onen--Roncal--Tapiola in the Euclidean setting, for this operator on $L^{p}(w)$. Comparing to the previous results in the Euclidean setting, our assumptions on the kernel and on the underlying space are weaker. Moreover, we investigate the quantitative weighted bound for the bi-parameter rough singular integrals on product homogeneous Lie groups.