V. Kokilashvili, S. Samko

Singular Integrals in Weighted Lebesgue Spaces with Variable Exponent

In the weighted Lebesgue space with variable exponent the boundedness of the Calderón-Zygmund operator is established. The variable exponent $p(x)$ is assumed to satisfy the logarithmic Dini condition and the exponent $\beta$ of the power weight $\rho(x)=|x-x_0|^{\beta}$ is related only to the value
$p(x_0)$. The mapping properties of Cauchy singular integrals defined on the Lyapunov curve and on curves of bounded rotation are also investigated within the framework of the above-mentioned weighted space.