Critical points of essential norms of singular integral operators in weighted spaces

We show that for any simple piecewise Ljapunov contour Gamma there exists a power weight rho such that the essential norm \S-Gamma\ in the space L-2(G amma, rho) does not depend on the angles of the contour and it is given by formula (2). All such weights are described. For the union Gamma = Gamma(1) boolean OR Gamma(2). Of two simple piecewise Lyapunov curves we prove that the essential norm \S-Gamma\ in L-2(Gamma) is minimal if both Gamma(1) and Gamma(2) are smooth in some neighborhoods of the common points. It is the case when the norm \S-Gamma\ in the space L-2(Gamma) as well as in L-2(Gamm a, rho) does not depend on the values of the angles and it can be calculate d by formula (5).