Hausdorff Operators on Compact Abelian Groups

TL;DR

This paper establishes unified boundedness conditions for Hausdorff operators across various function spaces on compact Abelian groups, with applications to Dirichlet series and continuous functions.

Contribution

It provides the first unified criteria for Hausdorff operator boundedness on multiple Hardy and BMO spaces over compact Abelian groups.

Findings

Conditions are the same for all spaces and groups considered.

Applications to Dirichlet series are demonstrated.

Includes analysis of continuous function spaces and examples.

Abstract

Necessary and sufficient conditions are given for boundedness of Hausdorff operators on generalized Hardy spaces $H^p_E(G)$, real Hardy space $H^1_{\mathbb{R}}(G)$, $BMO(G)$, 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.