Weighted norm inequalities for jump and variation of some singular integral operators with rough kernel

In this paper, we consider weighted jump and variational inequalities of truncated singular integral operator with rough kernel [Formula: see text] where the kernel [Formula: see text] satisfies the cancellation condition and the homogeneous condition of degree [Formula: see text] and [Formula: see text]. This kind of singular integral appears in the approximation of the surface quasi-geostrophic (SQG) equation from the generalized SQG equation. We establish the [Formula: see text] estimate of the jump and variational inequalities of the families [Formula: see text] for [Formula: see text]. Moreover, one can get the weighted [Formula: see text] boundedness of the Calderón–Zygmund operator by letting [Formula: see text].