<?xml version="1.0" encoding="UTF-8"?><xml><records><record><source-app name="Biblio" version="6.x">Drupal-Biblio</source-app><ref-type>5</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">Karlovich, Alexei Yu.</style></author><author><style face="normal" font="default" size="100%">Spitkovsky, Ilya M.</style></author></authors><secondary-authors><author><style face="normal" font="default" size="100%">Manuel Cepedello Boiso</style></author><author><style face="normal" font="default" size="100%">Håkan Hedenmalm</style></author><author><style face="normal" font="default" size="100%">Marinus A. Kaashoek</style></author><author><style face="normal" font="default" size="100%">Alfonso Montes Rodríguez</style></author><author><style face="normal" font="default" size="100%">Sergei Treil</style></author></secondary-authors></contributors><titles><title><style face="normal" font="default" size="100%">The Cauchy singular integral operator on weighted variable Lebesgue spaces</style></title><secondary-title><style face="normal" font="default" size="100%">Concrete Operators, Spectral Theory, Operators in Harmonic Analysis and Approximation. Operator Theory: Advances and Applications, 236</style></secondary-title></titles><dates><year><style  face="normal" font="default" size="100%">2014</style></year></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://link.springer.com/chapter/10.1007/978-3-0348-0648-0_17</style></url></web-urls></urls><publisher><style face="normal" font="default" size="100%">Birkhäuser</style></publisher><pub-location><style face="normal" font="default" size="100%">Basel</style></pub-location><pages><style face="normal" font="default" size="100%">275-291</style></pages><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;Let \(p:\mathbb{R}\to(1,\infty)\) be a globally log-Hölder continuous variable exponent and \(w:\mathbb{R}\to[0,\infty]\) be a weight. We prove that the Cauchy singular integral operator \(S\) is bounded on the weighted variable Lebesgue space \(L^{p(\cdot)}(\mathbb{R},w)=\{f:fw\in L^{p(\cdot)}(\mathbb{R})\}\) if and only if the weight \(w\) satisfies $$ \sup_{-\infty &amp;lt; a &amp;lt; b &amp;lt; \infty} \frac{1}{b-a} \|w\chi_{(a,b)}\|_{p(\cdot)} \|w^{-1}\chi_{(a,b)}\|_{p'(\cdot)}&amp;lt;\infty \quad (1/p(x)+1/p'(x)=1). $$
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