On extension of Calderón-Zygmund type singular integrals and their commutators

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 $$T_{\beta }f(x)=p.v.{\int }_{{\mathbb{R}}^n}\frac{\Omega (y)}{|y{|}^{n-\beta }}f(x-y)dy,$$ 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 Calderón-Zygmund type singular integrals and their commutators.

27 pages, submitted

References in corpus (1)

• Remarks on well-posedness of the generalized surface quasi-geostrophic equation