On boundedness of Hausdorff-type operators on Sobolev spaces

TL;DR

This paper introduces a new class of Hausdorff-type operators on function spaces and establishes a sharp sufficient condition for their boundedness on Sobolev spaces, highlighting the optimality of this condition.

Contribution

It proposes a novel Hausdorff-type operator concept and provides the first sharp boundedness criterion on Sobolev spaces, advancing operator theory in functional analysis.

Findings

Established a sufficient condition for boundedness

Proved the condition's optimality in general

Extended the understanding of Hausdorff-type operators

Abstract

A new notion of a Hausdorff-type operator on function spaces over domains in Euclidean spaces is introduced, and a sufficient condition for the boundedness of this operator on Sobolev spaces is proved. It is shown that this condition cannot be weakened in general.