The Plemelj–Privalov theorem for generalized Hölder classes

The Plemelj–Privalov theorem on invariance of the Hölder classes $H_\alpha$, $0<\alpha<1$, with respect to a one-dimensional singular integral with Cauchy kernel on closed smooth Jordan curves is well known. V. V. Salaev posed the problem of describing the class $\Pi_\alpha$ of all closed rectifiable Jordan curves on which the Plemelj–Privalov theorem is valid; it was solved in a paper by Salaev, Guseinov, and Seifullaev, where a condition completely characterizing the class $ \Pi_\alpha$ is given in terms of the planar measure of boundary strips of sets constructed from the curve. The present article is devoted to the solution of Salaev's problem for the generalized Hölder class $H_\varphi$.