Hausdorff operators on compact abelian groups

Abstract

Necessary and sufficient conditions are given for the boundedness of Hausdorff operators on the generalized Hardy spaces 𝐻 𝑝 𝐸 (𝐺), real Hardy space 𝐻 1 ℝ (𝐺), BMO(𝐺), and BMOA(𝐺) for compact Abelian group 𝐺. Surprisingly, these conditions turned out to be the same for all groups and spaces under consideration. Applications to Dirichlet series are given. The case of the space of continuous functions on 𝐺 and examples are also considered.

To the Spectral Theory of Discrete Hausdorff Operators

“…Let S = G be a topological group, Aut(S) = Aut(G) the group of all topological automorphisms of G. In this case, we have got a general definition of a Hausdorff operator on topological groups. This example contains several known definitions of Hausdorff operators for classical groups (see [24], [25], and examples therein). Let Ω = Z be endowed with the counting measure.…”