On the Derivatives of Cauchy-Type Integrals in the Polydisk

Abstract

In the paper the formulas for derivatives of the Cauchy type integral K[f] of smooth functions f on the distinguished boundary of a polydisc are given. These formulas express the derivatives of orderm from K[f] through the derivatives of lower order (Theorem 1). We use them for the estimates of smoothness of derivatives of Cauchy type integral in terms of the Holder’s ordinary scale (Theorem 2) and in terms of anisotropic Holder’s classes (Theorem 5).