Hausdorff operators on homogeneous spaces of locally compact groups

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.

Boundedness of Hausdorff operators on Hardy spaces over homogeneous spaces of Lie groups

“…The case of locally compact groups was considered earlier in [10]. The aim of this note is to improve and generalize results from [8] to the case of Hardy spaces H 1 (G/K) with the norm defined via (1, q) atoms when G is a Lie group and to apply results we obtain to generalized Delsarte operators and to Hausdorff operators over multidimensional spheres.…”

“…Definition 2. [8]. Let (Ω, µ) be a measure space, ( Ȧ(u)) u∈Ω ⊂ Aut K (G) a family of homeomorphisms of G/K, and Φ a measurable function on (Ω, µ).…”

“…Hausdorff operators on the Hardy space H 1 over homogeneous spaces of locally compact groups were first introduced by the author in [8] for the case of doubling measures, and in [9] for the case of locally doubling measures. The case of locally compact groups was considered earlier in [10].…”

Boundedness of Hausdorff operators on real Hardy spaces H1 over locally compact groups

“…In [6], [7] classical results on the boundedness of Hausdorff operators on the Hardy space H 1 over finite-dimensional real space were generalized to the case of a Hardy space over locally compact metrizable groups with the doubling property. In [16] Hausdorff operators on Lebesgue and real Hardy spaces over homogeneous spaces of locally compact groups with doubling property were considered. On the other hand, as was shown by T. Kawazoe, non compact semisimple Lie groups should not satisfy the doubling property but often enjoy the less restrictive so called local doubling property (see condition (LDP) below) [17,Lemma 2.6], [18, Lemma 3.2] 2 .…”

“…In [16] Hausdorff operators on the homogeneous space G/K were introduced in the following way. Recall that the quotient space G/K consists of left cosets ẋ := xK = π K (x) (x ∈ G) where π K : G → G/K stands for a natural projection.…”