Hölder estimates for the Cauchy integral on a Lipschitz contour
Let \(\Gamma\) be a Lipschitz contour. Note that a Lipschitz contour may have (infinitely many) corners, but may not posses cusps. The author considers the singular integral operator \[ (Su)(x)=u(x)+1/\pi i\int_{\Gamma}u(y)-u(x)/(y-x)dy,\quad x\in \Gamma, \] where \(u\in H^{\alpha}(\Gamma)\), \(0<\alpha <1\), and estimates the norm of the operator \(S\in B(H^{\alpha}(\Gamma))\). Moreover, a theorem analogous to the Plemelj-Sokhotskij theorem for the integral 1/2\(\pi\) \(i\int_{\Gamma}u(y)dy/(y-z)\) is obtained.
scientific article; zbMATH DE number 4103713