Hausdorff Operators on Compact Abelian Groups

Abstract

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