Quantitative weighted estimates for rough singular integrals on homogeneous groups
Zhijie Fan, Ji Li

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 boundedness ( and a Muckenhoupt weight) of singular integrals with homogeneous convolution kernel on an arbitrary homogeneous group of dimension , {under the assumption that , the restriction of to the unit annulus, is mean zero and integrable for some ,} where is a fixed constant depending on . 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 . 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.
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Taxonomy
TopicsAdvanced Harmonic Analysis Research · Advanced Mathematical Physics Problems · Mathematical Analysis and Transform Methods
