Singular integral operators with the complex conjugation on curves with cusps (Q1892630)

The authors investigate on the weighted Lebesgue space \(RL^m_p(\Gamma, \rho)\) of vector-functions over the field of real numbers operators in the Banach algebra generated by the Cauchy singular integral operator \[ S_\Gamma\varphi(t)= {1\over \pi i} \int_\Gamma {\varphi(\tau) d\tau\over \tau- t}, \] the operator of the complex conjugation and the operators of multiplication by \(m\times m\) matrix- functions continuous on \(\Gamma\backslash \{c_1,\dots, c_n\}\) which may have jumps at knots \(c_1,\dots, c_n\in \Gamma\), where \(\Gamma\) is a closed, oriented, simple, piecewise-smooth Lyapunov curve with the angle \(\gamma_j\) at \(c_j\). The angles \(\gamma_j= 0\) and \(\gamma_j= 2\pi\) are admissible and in that case the curve is said to have a cusp at \(c_j\). The notion of order of the cusp is defined. The main results of the paper include necessary and sufficient conditions for the above-mentioned operators to be Fredholm ones in the case where all cusps of \(\Gamma\) have the order 1.