The Cauchy Singular Integral Operator on Weighted Variable Lebesgue Spaces

Abstract

Let p: ℝ → (1,∞) be a globally log-Hölder continuous variable exponent and w: ℝ →[0,∞] be a weight. We prove that the Cauchy singular integral operator s is bounded on the weighted variable Lebesgue space L ^p(.)(ℝ,w)= {f:f wϵL ^p(.)(ℝ)} if and only if the weight w satisfies

$\mathop{sup}\limits_{-\infty<a<b<\infty}\frac{1}{b-a}\parallel w\chi_{(a,b)}\parallel_{p(.)}\parallel w^{-1}\chi_{(a,b)}\parallel_{p^{\prime}(.)}<\infty\quad (1/p(x)+1/p^{\prime}(x)=1).$

Mathematics Subject Classification (2010). Primary 42A50; Secondary 42B25, 46E30.