Boundedness of one class of Volterra-type integral operators in Lebesgue spaces and their applications

We consider expanding classes of kernels of integral operators which include, as a special case, some well-known classes of kernels. We establish various properties of these classes. We obtain boundedness criteria for Volterra-type integral operators in Lebesgue spaces with a parameter when the kernels of the Volterra operator belong to the expanding classes. As an application of the obtained results, we study weighted estimates for quasilinear integral operators containing iterations of two Volterra-type integral operators with kernels from the expanding classes.