On the norms of singular integral operators on contours with intersections (Q1042640)

The following singular integral operator \(S_{\Gamma}\) with Cauchy kernel is considered \[ S_{\Gamma}f(t)=\frac{1}{\pi i}\int\limits_{\Gamma}f(\tau)\frac{d\tau}{\tau-t}. \] It is proved that \(S_{\Gamma}\) is bounded on the spaces \(L_p(\Gamma),\; 1<p<\infty\). The exact bounds are obtained for the norm of this operator on the space \(L_2(\Gamma)\) on the family of rays originating at the same point. These bounds, with the use of the localization technique, are then extended to the essential norm \(|S_{\Gamma}|\) on piecewise smooth curves with finitely many points of self-intersection.