An extension of Calderón-Zygmund type singular integral with non-smooth kernel

Abstract In the present paper, we consider a kind of singular integral T f ( x ) = p . v . ∫ R n Ω ( y ) | y | n − β f ( x − y ) d y which can be viewed as an extension of the classical Calderon-Zygmund type singular integral. This kind of singular integral appears in the approximation of the surface quasi-geostrophic (SQG) equation from the generalized SQG equation. We establish an estimate of the singular integral in the L q space for 1 q ∞ and a weak ( 1 , 1 ) type of the singular integral when 0 β ( q − 1 ) n q without any smoothness assumed on Ω. Moreover, the bounds do not depend on β and the strong ( q , q ) type estimate and weak ( 1 , 1 ) type estimate of the Calderon-Zygmund type singular integral can be recovered when β → 0 from our obtained estimates.