On weighted estimates for a class of Volterra integral operators

Volterra integral operators ${cal A}=sum_{k=0}^m {cal A}_k$, $({cal A}_k f)(x)= a_k (x)int_0^x t^k f(t) ,dt$, are studied acting between weighted $L_2$ spaces on $(0,+infty)$. Under certain conditions on the weights and functions $a_k$, it is shown that $cal A$ is bounded if and only if each ${cal A}_k$ is bounded. This result is then applied to describe spaces of pointwise multipliers in weighted Sobolev spaces on $(0,+infty)$.