<?xml version="1.0" encoding="UTF-8"?><xml><records><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">On the essential norm of the Cauchy singular integral operator in weighted rearrangement-invariant spaces</style></title><secondary-title><style face="normal" font="default" size="100%">Integral Equations and Operator Theory</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2000</style></year><pub-dates><date><style  face="normal" font="default" size="100%">{SEP}</style></date></pub-dates></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://link.springer.com/article/10.1007/BF01192300</style></url></web-urls></urls><number><style face="normal" font="default" size="100%">{1}</style></number><publisher><style face="normal" font="default" size="100%">{BIRKHAUSER VERLAG AG}</style></publisher><pub-location><style face="normal" font="default" size="100%">{VIADUKSTRASSE 40-44, PO BOX 133, CH-4010 BASEL, SWITZERLAND}</style></pub-location><volume><style face="normal" font="default" size="100%">38</style></volume><pages><style face="normal" font="default" size="100%">28-50</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">&lt;script src='https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML'&gt;&lt;/script&gt;&lt;p&gt;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 \(\Gamma\):&lt;br /&gt;
\[&lt;br /&gt;
|S|\ge\cot(\pi\lambda_{\Gamma,w}/2)&lt;br /&gt;
\]&lt;br /&gt;
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 \(\beta(Q_tw)\). &lt;/p&gt;
</style></abstract><work-type><style face="normal" font="default" size="100%">{Article}</style></work-type></record></records></xml>