Singular integrals and potentials in some Banach function spaces with variable exponent

We introduce a new Banach function space - a Lorentz type space with variable exponent. In this space the boundedness of singular integral and potential type operators is established, including the weighted case. The variable exponent p(t) is assumed to satisfy the logarithmic Dini condition and the exponent β of the power weight ω(t)=|t|β is related only to the value p(0). The mapping properties of Cauchy singular integrals defined on Lyapunov curves and on curves of bounded rotation are also investigated within the framework of the introduced spaces.