Boundedness in Lebesgue Spaces with Variable Exponent of the Cauchy Singular Operator on Carleson Curves

Abstract

We prove the boundedness of the singular integral operator S Γ in the spaces L p(·)(Γ, ρ) with variable exponent p(t) and power weight ρ on an arbitrary Carleson curve under the assumptions that p(t) satisfy the logcondition on Γ. The curve Γ may be finite or infinite.We also prove that if the singular operator is bounded in the space L p(·)(Γ), then Γ is necessarily a Carleson curve. A necessary condition is also obtained for an arbitrary continuous coefficient.KeywordsWeighted generalized Lebesgue spacesvariable exponentsingular operatorCarleson curves