@inbook {6700,
	title = {The Cauchy singular integral operator on weighted variable Lebesgue spaces},
	booktitle = {Concrete Operators, Spectral Theory, Operators in Harmonic Analysis and Approximation. Operator Theory: Advances and Applications, 236},
	year = {2014},
	pages = {275-291},
	publisher = {Birkh{\"a}user},
	organization = {Birkh{\"a}user},
	address = {Basel},
	abstract = {<script src={\textquoteright}https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML{\textquoteright}></script><p>Let \(p:\mathbb{R}\to(1,\infty)\) be a globally log-H{\"o}lder 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 $$ \sup_{-\infty \&lt; a \&lt; b \&lt; \infty} \frac{1}{b-a} \|w\chi_{(a,b)}\|_{p(\cdot)} \|w^{-1}\chi_{(a,b)}\|_{p{\textquoteright}(\cdot)}\&lt;\infty \quad (1/p(x)+1/p{\textquoteright}(x)=1). $$
</p>
},
	url = {http://link.springer.com/chapter/10.1007/978-3-0348-0648-0_17},
	author = {Karlovich, Alexei Yu. and Spitkovsky, Ilya M.},
	editor = {Manuel Cepedello Boiso and H{\r a}kan Hedenmalm and Marinus A. Kaashoek and Alfonso Montes Rodr{\'\i}guez and Sergei Treil}
}
