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

Defining Hausdorff operators on Euclidean spaces

“…In the related topic of the compactness (or noncompactness) of the Hausdorff operators, only first results have appeared, see other Mirotin. [2][3][4] There is much to do in this topic; open problems are also discussed in Liflyand. 7,22…”

“…There are more abstract settings as well; see Mirotin. [2][3][4] In this work, we are mainly concentrated on the Hausdorff operators in the Euclidean space R n . The most general form in which the Hausdorff operators have been considered in the Euclidean setting until recently is ( )(x) = ( Φ )(x) = ( Φ,A )(x) = ∫ R n Φ(u) (xA(u)) du,…”

On the Structure of Normal Hausdorff Operators on Lebesgue Spaces

“…Below we shall shaw that such an operator is noncompact in L p (R n ) (1 ≤ p < ∞) if (A(u)) u∈Ω is a pairwise commuting family of positive definite matrices. The case p = 2 was considered in [13].…”

“…Definition 1 [13]. Let (Ω, µ) be some σ-compact locally compact topological space endowed with positive regular Borel measure µ, Φ a locally integrable function on Ω, and (A(u)) u∈Ω a µ-measurable family of n × n matrices defined almost everywhere in the support of Φ and satisfying detA(u) = 0.…”

Hausdorff operators on homogeneous spaces of locally compact groups

“…Hausdorff operators on general groups were defined and studied by the author in [8] and [9]. 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.…”