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.
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Kokilashvili, V., Paatashvili, V., Samko, S. (2006). Boundedness in Lebesgue Spaces with Variable Exponent of the Cauchy Singular Operator on Carleson Curves. In: Erusalimsky, Y.M., Gohberg, I., Grudsky, S.M., Rabinovich, V., Vasilevski, N. (eds) Modern Operator Theory and Applications. Operator Theory: Advances and Applications, vol 170. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-7737-3_10
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