On extension of Calderón-Zygmund type singular integrals and their commutators
Sayan Bagchi, Rahul Garg, Joydwip Singh
TL;DR
We address extending Calderón–Zygmund type singular integrals by introducing $T_\beta$ with kernel $\Omega(y)/|y|^{n-\beta}$ for $0<β<n$ and study the boundedness of $T_\beta$ and its commutator with $b$ across Lipschitz, Hardy and $A_p$ weighted spaces. The main approach combines a two-term kernel decomposition $T_\beta=T_1+T_2$, Hardy space–Lipschitz duality via atoms and molecules, and sharp maximal function techniques to obtain bounds that are uniform in small $β$, enabling the limit $β\to0$ to recover the classical CZ theory and CRW-type commutators under a Dini condition on $Ω$. The paper also yields weighted $L^p(ω)$ and Hardy space estimates for the commutators, with explicit $β$-dependent terms, and extends the analysis to Lipschitz and Hardy space mapping properties. These results provide quantitative control of the $β$-dependent extension and have potential applications in PDEs such as generalized SQG and regularity questions where approximation by $T_β$ is relevant.
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 Calderón-Zygmund type singular integrals $$ T_βf (x) = p.v. \int_{\mathbb{R}^n} \frac{Ω(y)}{|y|^{n-β}} f(x-y) \, dy, $$ for $0 < β< n$, 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 $β$, and therefore one can pass onto the limits as $β\to 0$ to deduce analogous estimates for the classical Calderón-Zygmund type singular integrals and their commutators.