Singular integrals generated by zonal measures (Q2759023)

We study \(L^p\)-mapping properties of the rough singular integral operator NEWLINE\[NEWLINET_\nu f(x)=\int_0^\infty dr/r \int_{\Sigma_{n - 1}} f(x-r\theta)d\nu(\theta)NEWLINE\]NEWLINE 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.