Boundedness of a class of super singular integral operators and the associated commutators

AbstractIn this paper we give the (Lpα, Lp) boundedness of the maximal operator of a class of super singular integrals defined by $$T_{\Omega ,\alpha }^* f(x) = \mathop {\sup }\limits_{\varepsilon > 0} \left| {\int_{|x - y| > \varepsilon } {b(|y|)} \Omega (y)|y|^{ - n - \alpha } f(x - y)dy} \right|,$$ which improves and extends the known result. Moreover, by applying an off-Diagonal T1 Theorem, we also obtain the (Lp, Lq) boundedness of the commutator defined by $$C_{\Omega ,\alpha } f(x) = p.v. \int_{\mathbb{R}^n } {(A(x)} - A(y))\Omega (x - y)|x - y|^{ - n - \alpha } f(y)dy.$$