On the essential norm of the Cauchy singular integral operator in weighted rearrangement-invariant spaces

In this paper we extend necessary conditions for Fredholmness of singular integral operators with piecewise continuous coefficients in rearrangement-invariant spaces [19] to the weighted case X(Gamma,w). These conditions are formulated in terms of indices alpha(Q(t)w) and beta(Q(t)w) of a submultiplicative function Q(t)w, which is associated with local properties of the space, of the curve, and of the weight st the point t is an element of Gamma. Using these results we obtain a lower estimate for the essential norm IS! of the Cauchy singular integral operator S in reflexive weighted rearrangement-invariant spaces X(Gamma, w) over arbitrary Carleson curves Gamma: \S\ greater than or equal to cot (pi lambda(Gamma,w)/2) where lambda(Gamma,w) :=inf/t is an element of Gamma min{alpha(Q(t)w), 1 - beta(Q(t)w)}. In some cases we give formulas for computation of alpha(Q(t)w) and beta(Q(t)w).