Singular integrals generated by zonal measures

We study $L^p$-mapping properties of the rough singular integral operator $T_\nu f(x)=\int_0^\infty dr/r \int_{\Sigma_{n - 1}} f(x-r\theta)d\nu(\theta)$ depending on a finite Borel measure $\nu$ on the unit sphere $\Sigma_{n -1}$ in $\mathbb{R}^n$. It is shown that the conditions $\sup_{\vert\xi \vert=1} \int_{\Sigma_{n -1}} \log \;(1/\vert \theta \cdot \xi \vert) d\vert\nu\vert(\theta) < \infty$, $\nu(\Sigma_{n - 1})=0$ imply the $L^p$-boundedness of $T_\nu$ for all $1<p<\infty$ provided that $n>2$ and $\nu$ is zonal.