  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.
