Singular integral operators with a generalized Cauchy kernel

Singular integral operators with piecewise continuous matrix coefficients are considered on a piecewise smooth curve in weighted Lebesgue spaces. In contrast to the classical case, the operators have generalized Cauchy kernels arising as a parametrix of first-order elliptic systems in the plane. A Fredholmness criterion and an index formula for these operators are obtained in weighted Lebesgue spaces.