Hausdorff Operators on Compact Abelian Groups

Abstract

Necessary and sufficient conditions are given for boundedness of Hausdorff operators on generalized Hardy spaces HpE(G), real Hardy space H1R(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.