Singular integrals generated by zonal measures

Abstract:

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 $\sup _{|\xi |=1} \int _{\Sigma _{n -1}} \log \;(1/| \theta \cdot \xi |) d|\nu |(\theta ) < \infty$, $\nu (\Sigma _{n - 1})=0$ imply the $L^p$-boundedness of $T_\nu$ for all $1<p<\infty$ provided that $n>2$ and $\nu$ is zonal.