On a general concept of a Hausdorff-type operator

TL;DR

This paper introduces a unified framework for Hausdorff-type operators, establishing boundedness conditions in various function spaces and exploring their regularity, aiming to unify the study of classical and new operators.

Contribution

It proposes a general definition encompassing classical and new Hausdorff operators, providing boundedness criteria and regularity analysis within a unified approach.

Findings

Boundedness conditions in $L^p$ and $H^1$ spaces.

Regularity properties of the operators.

Examples illustrating the unified framework.

Abstract

A unified approach to the concept of a Hausdorff operator is proposed in such a way that a number of classical and new operators feet into the given definition. Conditions are given for the boundedness of the operators under consideration in $L^p$ and in the atomic Hardy space $H^1$, and their regularity property is investigated. Examples are considered. The author hopes that this approach will allow one to unify the study of a lot of extensions and analogs of the classical Hausdorff operator.