## The Cauchy singular integral operator on weighted variable Lebesgue spaces

Citation:
Karlovich, Alexei Yu., and Ilya M. Spitkovsky. "The Cauchy singular integral operator on weighted variable Lebesgue spaces." Concrete Operators, Spectral Theory, Operators in Harmonic Analysis and Approximation. Operator Theory: Advances and Applications, 236. Eds. Manuel Cepedello Boiso, Håkan Hedenmalm, Marinus A. Kaashoek, Alfonso Montes Rodríguez, and Sergei Treil. Basel: Birkhäuser, 2014. 275-291.

### Abstract:

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 $$\sup_{-\infty < a < b < \infty} \frac{1}{b-a} \|w\chi_{(a,b)}\|_{p(\cdot)} \|w^{-1}\chi_{(a,b)}\|_{p'(\cdot)}<\infty \quad (1/p(x)+1/p'(x)=1).$$