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 $\Hp^p$, $p \geq 1$, have been the focus of increasing interest among researchers following a paper of Hedenmalm, Lindqvist and Seip in Duke Math. J 86 (1997), 1-37. The Dirichlet series in these spaces converge on a certain half-plane, where one may also define the classical Hardy spaces $H^p$. In this paper, we compare the boundary behaviour of elements in $\Hp^p$ and $H^p$. Moreover, Carleson measures of the spaces $\Hp^p$ 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 possible to find a function in the corresponding Dirichlet-Hardy space such that their difference has an analytic continuation across this interval.