Jump and Variational Inequalities for Singular Integral with Rough Kernel

In this paper, we consider the jump and variational inequalities of truncated singular integral operator with rough kernel

$$T_{\Omega,\beta,\varepsilon}f(x)=\int_{\mid y\mid>\varepsilon}{\Omega(y)\over \mid y\mid ^{n-\beta}}f(x-y)dy,$$

where the kernel \(\Omega \in (L(\log^{+}L)^{2})^{n \over{n-\beta}}(\mathbb{S}^{n-1})\) satisfies the vanishing condition and the homogeneous condition of degree 0. This kind of singular integral appears in the approximation of the surface quasi-geostrophic (SQG) equation from the generalized SQG equation. We establish the (L^p, L^q) estimate of the jump and variational inequalities of the families {T_Ω,β,ε}_ε>0 for \({1\over q}={1\over p}-{\beta\over n}\) and 0 < β < 1. Moreover, one can get the L^p boundedness of the Calderón–Zygmund operator with the same kernel by letting β → 0⁺.