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 Lp(⋅)(ℝ,w) = {f : fw ∈ Lp(⋅)(ℝ)} if and only if the weight w satisfies (Formula Presented).