**V. Kokilashvili, S. Samko**

## Singular Integrals in Weighted
Lebesgue Spaces with Variable Exponent

**abstract:**

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.