# The Cauchy singular integral operator on weighted variable Lebesgue spaces

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## Abstract

Let $p:\R\to(1,\infty)$ be a globally log-H\"older continuous variable exponentand $w:\R\to[0,\infty]$ be a weight. We prove that the Cauchy singular integraloperator $S$ is bounded on the weighted variable Lebesgue space$L^{p(\cdot)}(\R,w)=\{f:fw\in L^{p(\cdot)}(\R)\}$ if and only if the weight $w$satisfies$\sup_{-\infty<a<b<\infty}\frac{1}{b-a}\|w\chi_{(a,b)}\|_{p(\cdot)}\|w^{-1}\chi_{(a,b)}\|_{p'(\cdot)}<\infty\quad (1/p(x)+1/p'(x)=1).$
Original language Unknown Operator Theory: Advances and Applications 275-291 978-3-0348-0648-0 https://doi.org/10.1007/978-3-0348-0648-0_17 Published - 1 Jan 2014 Concrete Operators, Spectral Theory, Operators in Harmonic Analysis and Approximation. 22nd International Workshop in Operator Theory and its Applications, Sevilla, July 2011 - Duration: 1 Jan 2011 → …

### Conference

Conference Concrete Operators, Spectral Theory, Operators in Harmonic Analysis and Approximation. 22nd International Workshop in Operator Theory and its Applications, Sevilla, July 2011 1/01/11 → …