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

Authors:

Email author" target="_blank">Qionglei?Chen Email author Zhifei?Zhang

Institution:

1. Department of Mathematics, Zhejiang University, Hangzhou 310028, China
2. Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing 100080, China

Abstract:

In this paper we give the (L ^p _α, L ^p) 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 (L ^p, L ^q) boundedness of the commutator defined by

$C_{\Omega ,\alpha } f(x) = p.v. \int_{\mathbb{R}^n } {(A(x)} - A(y))\Omega (x - y)|x - y|^{ - n - \alpha } f(y)dy.$

Keywords:

singular integral operator maximal operator commutator Tl Theorem

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