An algebra of integral operators with fixed singularities in kernels

We continue the study of algebras generated by the Cauchy singular integral operator and integral operators with fixed singularities on the unit inter val, started in R.Duduchava, E.Shargorodsky, 1990. Such algebras emerge whe n one considers singular integral operators with complex conjugation on cur ves with cusps. As one of possible applications of the obtained results we find an explicit formula for the local norms of the Cauchy singular integra l operator on the Lebesgue space L-2(Gamma, rho), where Gamma is a curve wi th cusps of arbitrary order and rho is a power weight. For curves with angl es and cusps of order 1 the formula was already known (see R.Avedanio, N.Kr upnik, 1988 and R.Duduchava, N.Krupnik, 1995).