Quantitative weighted estimates for rough singular integrals on homogeneous groups

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 $q_0<q\le \infty$,} where $q_0$ is a fixed constant depending on $w$. We obtain a quantitative weighted bound, which is consistent with the one obtained by Hytönen--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.

arXiv comment29 pages, typos fixed

References in corpus (6)

• $L^p$ bounds for singular integrals and maximal singular integrals with rough kernels
• On pointwise estimates involving sparse operators
• A sparse domination principle for rough singular integrals
• Sparse Domination for Bi-Parameter Operators Using Square Functions
• $A_1$ theory of weights for rough homogeneous singular integrals and commutators
• Modern singular integral theory with mild kernel regularity

Cited by in corpus (1)

• $L^p$ estimates and weighted estimates of fractional maximal rough singular integrals on homogeneous groups