On the boundary behaviour of the Hardy spaces of Dirichlet series and a frame bound estimate

Abstract

A range of Hardy-like spaces of ordinary Dirichlet series, called the Dirichlet–Hardy spaces ℋp, p ≧ 1, have been the focus of increasing interest among researchers following a paper of Hedenmalm, Lindqvist and Seip [16]. The Dirichlet series in these spaces converge on a certain half-plane, where one may also define the classical Hardy spaces Hp. In this paper, we compare the boundary behaviour of elements in ℋp and Hp. Moreover, Carleson measures for the spaces ℋp are studied. Our main result shows that for certain cases the following statement holds true. Given an interval on the boundary of the half-plane of definition and a function in the classical Hardy space, it is possible to find a function in the corresponding Dirichlet–Hardy space such that their difference has an analytic continuation across this interval.