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 caseX(Γ,w). These conditions are formulated in terms of indices α(Qtw) and β(Qtw) of a submultiplicative functionQtw, 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:\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\left| S \right| \geqslant \cot \left( {\pi \lambda _\Gamma ,w/2} \right)$$ \end{document} where\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\lambda _{\Gamma ,w} : = \begin{array}{*{20}c} {\inf } \\ {t \in \Gamma } \\ \end{array} \min \left\{ {\alpha \left( {Q_t w} \right),1 - \beta \left( {Q_t w} \right)} \right\}$$ \end{document}. In some cases we give formulas for computation of α(Qtw) and β(Qtw).