@article { ISI:000089786400003, title = {On the essential norm of the Cauchy singular integral operator in weighted rearrangement-invariant spaces}, journal = {Integral Equations and Operator Theory}, volume = {38}, number = {{1}}, year = {2000}, month = {{SEP}}, pages = {28-50}, publisher = {{BIRKHAUSER VERLAG AG}}, type = {{Article}}, address = {{VIADUKSTRASSE 40-44, PO BOX 133, CH-4010 BASEL, SWITZERLAND}}, abstract = {

In this paper we extend necessary conditions for Fredholmness of singular integral operators with piecewise continuous coefficients in rearrangement-invariant spaces [K98] to the weighted case \(X(\Gamma,w)\). These conditions are formulated in terms of indices \(\alpha(Q_tw)\) and \(\beta(Q_tw)\) of a submultiplicative function \(Q_tw\), which is associated with local properties of the space, of the curve, and of the weight at the point \(t\Gamma\). Using these results we obtain a lower estimate for the essential norm \(S\) of the Cauchy singular integral operator \(S\) in reflexive weighted rearrangement-invariant spaces \(X(\Gamma, w)\) over arbitrary Carleson curves \(\Gamma\):
\[
|S|\ge\cot(\pi\lambda_{\Gamma,w}/2)
\]
where \(\lambda_{\Gamma,w} :=inf_{t\in\Gamma} min\{\alpha(Q_tw), 1 - \beta(Q_tw)\}\). In some cases we give formulas for computation of \(\alpha(Q_tw)\) and \(\beta(Q_tw)\).

}, issn = {{0378-620X}}, doi = {{10.1007/BF01192300}}, url = {http://link.springer.com/article/10.1007/BF01192300}, author = {Karlovich, Alexei Yu.} }