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 T β f (x) = p. v. ∫ R n Ω (y) | y | n - β f (x - y) d y , <graphic href="12044_2024_804_Article_Equ26.gif"></graphic> 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 → 0 to deduce analogous estimates for the classical Calderón–Zygmund type singular integrals and their commutators. [ABSTRACT FROM AUTHOR]