The Cauchy Singular Integral Operator on Weighted Variable Lebesgue Spaces

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

17 pages