On weighted estimates for a class of Volterra integral operators

TL;DR

This paper investigates the boundedness of a class of weighted Volterra integral operators on weighted $L_2$ spaces, establishing conditions under which the sum of operators is bounded based on individual components, with applications to weighted Sobolev spaces.

Contribution

It provides new necessary and sufficient conditions for the boundedness of weighted Volterra operators and applies these results to characterize pointwise multipliers in weighted Sobolev spaces.

Findings

Boundedness of the sum of Volterra operators is equivalent to boundedness of each component.

Conditions on weights and functions ensure operator boundedness.

Application to pointwise multipliers in weighted Sobolev spaces.

Abstract

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 $(0,+\infty)$.