Direct and inverse estimates for a singular Cauchy integral along a closed curve

A new metric characteristic $\theta(\delta)$ of rectifiable Jordan curves is introduced. We will find an estimate of the type of the Zygmund estimate for an arbitrary rectifiable closed Jordan curve in its terms. It is shown that the Plemel'–Privalov theorem on the invariance of Holder's spaces is true for the class of curves satisfying the condition $\theta(\delta)\sim\delta$, which is much wider than the class of piecewise smooth curves (the presence of cusps is admissible). The Bari–Stechkin theorem on the necessary conditions of action of a singular operator in the spaces $H_\omega$ is generalized. It is shown that this theorem is valid for every curve which has a continuous tangent at least at one point and $\theta(\delta)\sim\delta$.