On extension of Calder\'on-Zygmund type singular integrals and their commutators

TL;DR

This paper extends Calderón-Zygmund singular integrals with a parameter, establishing uniform estimates across various function spaces, and analyzes their commutators, connecting to classical integrals as the parameter approaches zero.

Contribution

It introduces a parameterized extension of Calderón-Zygmund integrals, providing uniform estimates and analyzing their commutators on multiple function spaces, bridging to classical results.

Findings

Uniform estimates for extended singular integrals on Lipschitz, Hardy, and weighted spaces.

Estimates for commutators of these integrals on Lebesgue and Hardy spaces.

Convergence of estimates to classical Calderón-Zygmund integrals as the parameter tends to zero.

Abstract

Motivated by the recent works [Huan Yu, Quansen Jiu, and Dongsheng Li, 2021] and [Yanping Chen and Zihua Guo, 2021], we study the following extension of Calder\'on-Zygmund type singular integrals $$ T_{\beta}f (x) = p.v. \int_{\mathbb{R}^n} \frac{\Omega(y)}{|y|^{n-\beta}} f(x-y) \, dy, $$ for $0 < \beta < n$, and their commutators. We establish estimates of these singular integrals on Lipschitz spaces, Hardy spaces and Muckenhoupt $A_p$-weighted $L^p$-spaces. We also establish Lebesgue and Hardy space estimates of their commutators. Our estimates are uniform in small $\beta$, and therefore one can pass onto the limits as $\beta \to 0$ to deduce analogous estimates for the classical Calder\'on-Zygmund type singular integrals and their commutators.