Singular integral operators with generalized Cauchy kernel on piecewise smooth contour

Singular integral operators of two types are considered on a piecewise smooth contour in weighted Lebesgue spaces with generalized Cauchy kernels, related to the parametrix of elliptic systems of first order on the plane. The operators of the first type are linear over the field of complex numbers and are represented as a usual combination of the generalized singular Cauchy operator and the operators of multiplication by piecewise continuous matrix functions. The operators of the second type act in the space of real vector functions and, thus, are linear over $\mathbb{R}$. They arise in the direct reduction of elliptic boundary problems using integral representations. A criterion is obtained for these operators to be Fredholm, and a formula for their index is indicated.