On two-dimensional singular integral operators with conformal Carleman shift

For the class of singular integral operators with continuous coefficients and with the conformal shift over a two-dimensional bounded domain $G \subset \mathbb C$ an explicit Fredholm property criterion is obtained. Operators under consideration have kernels $[(\bar \varsigma - \bar z)/(\varsigma - z)]^k \left| {\varsigma - z} \right|^{ - 2}$ either with positive or with negative $k \in \mathbb Z\backslash \{0\}$; the conformal shift $W\varphi (z) = \varphi (\omega (z))$, $\omega : G \to G$ is of Carleman type: $W^k
e I$ for k = 1, 2, ..., n – 1 and W^n = I. It is proved also that a Fredholm operator A of such type has trivial index Ind A = 0