On Hausdorff operators on homogeneous spaces of locally compact groups

TL;DR

This paper extends the study of Hausdorff operators to Lebesgue and Hardy spaces on homogeneous spaces of locally compact groups, introducing atomic Hardy spaces and establishing boundedness conditions.

Contribution

It defines Hausdorff operators on these spaces and introduces atomic Hardy spaces, providing new boundedness criteria and exploring their properties on homogeneous groups.

Findings

Boundedness conditions for Hausdorff operators established.

Introduction of atomic Hardy spaces on homogeneous spaces.

Several corollaries and open problems discussed.

Abstract

Hausdorff operators on the real line and multidimensional Euclidean spaces originated from some classical summation methods. Now it is an active research area. Hausdorff operators on general groups were defined and studied by the author since 2019. The purpose of this paper is to define and study Hausdorff operators on Lebesgue and real Hardy spaces over homogeneous spaces of locally compact groups. We introduce in particular an atomic Hardy space over homogeneous spaces of locally compact groups and obtain conditions for boundedness of Hausdorff operators on such spaces. Several corollaries are considered and unsolved problems are formulated.