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

Motivated by the recent works [1, 23], we study the following extension of Calderón–Zygmund type singular integrals \(\begin{aligned} T_{\beta }f (x) = p.v. \int _{\mathbb {R}^n} \frac{\Omega (y)}{|y|^{n-\beta }} f(x-y) \, dy, \end{aligned}\) $$\begin{array}{c}\hfill T_{\beta }f(x)=p.v.{\int }_{{\mathbb{R}}^n}\frac{\mathrm{\Omega }(y)}{{|y|}^{n-\beta }}f(x-y)dy,\end{array}$$ for \(0< \beta < n\) $0<\beta <n$ , and their commutators. We establish estimates of these singular integrals on Lipschitz spaces, Hardy spaces and Muckenhoupt \(A_p\) $A_p$ -weighted \(L^p\) $L^p$ -spaces. We also establish Lebesgue and Hardy space estimates of their commutators. Our estimates are uniform in small \(\beta \) $\beta$ , and therefore one can pass onto the limits as \(\beta \rightarrow 0\) $\beta \to 0$ to deduce analogous estimates for the classical Calderón–Zygmund type singular integrals and their commutators.