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

Abstract

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 caseX(Γ,w). These conditions are formulated in terms of indices α(Q t w) and β(Q t w) of a submultiplicative functionQ t w, which is associated with local properties of the space, of the curve, and of the weight at the pointt∈Γ. Using these results we obtain a lower estimate for the essential norm |S| of the Cauchy singular integral operatorS in reflexive weighted rearrangement-invariant spacesX(Γ,w) over arbitrary Carleson curves Γ: $$\left| S \right| \geqslant \cot \left( {\pi \lambda _\Gamma ,w/2} \right)$$ where\(\lambda _{\Gamma ,w} : = \begin{array}{*{20}c} {\inf } \\ {t \in \Gamma } \\ \end{array} \min \left\{ {\alpha \left( {Q_t w} \right),1 - \beta \left( {Q_t w} \right)} \right\}\). In some cases we give formulas for computation of α(Q t w) and β(Q t w).