On the smoothness of Cauchy type integrals

E. M. Dyn'kin

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part IX, Tome 92 (1979), pp. 115-133

Résumé

Let $f$ be a continuous function on a closed rectifiable Jordan curve $\Gamma$, $Kf$ its Cauchy integral (Considered as a function analytic in the Jordan region with the boundary $\Gamma$). The article deals with estimates of the smoothness of $Kf$ in $\Bar G$ in terms of moduli of smoothness of $f$. Principal results for the Hölder–Zygmund classes $\bigwedge^\alpha$ are as follows: a) $K[\bigwedge^\alpha(\Gamma)]\subset\bigwedge^\alpha(\Bar G)$ ($\alpha\ge1$); b)$K[\bigwedge^\alpha(\Gamma)]\subset\bigwedge^{2\alpha-1}(\Bar G)$ ($1/2\alpha1$); c) there is a $\Gamma$ and an $f\in\bigwedge^\frac12(\Gamma)$ such that $\sup_G|Kf|=+\infty$; d) for every $\beta\in(\max(0,2\alpha-1),d)$ there is a pair ($\Gamma f)$ such that $f\in\bigwedge^\alpha(\Gamma)$, $K[\bigwedge^\alpha(\Gamma)]\subset\bigwedge^\alpha(\Bar G)$, $\omega_G(Kf,\delta)\ge\operatorname{const}\cdot\delta^\beta$ ($\omega_G$ being the continuity modaley). A precise sufficient condition of the continuity of $KF$ in $\Bar G$ (expressed in terms of $\omega_\Gamma(f))$ is given.