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

We show that for any simple piecewise Ljapunov contour Γ there exists a power weight ρ such that the essential norm |SΓ| in the spaceL2(Γ, ρ) does not depend on the angles of the contour and it is given by formula (2). All such weights are described. For the union Γ=Γ1∪Γ2 of two simple piecewise Lyapunov curves we prove that the essential norm |SΓ| inL2(Γ) is minimal if both Γ1 and Γ2 are smooth in some neighborhoods of the common points. It is the case when the norm |SΓ| in the spaceL2(Γ) as well as inL2(Γ, ρ) does not depend on the values of the angles and it can be calculated by formula (5).