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

TL;DR

This paper investigates the boundary behavior of Dirichlet-Hardy spaces of Dirichlet series, compares them with classical Hardy spaces, and studies Carleson measures, revealing how functions in these spaces relate across boundary intervals.

Contribution

It provides new insights into the boundary behavior of Dirichlet-Hardy spaces and establishes a connection with classical Hardy spaces through analytic continuation results.

Findings

Boundary behavior of Dirichlet-Hardy spaces compared to classical Hardy spaces

Existence of functions in Dirichlet-Hardy spaces matching classical Hardy functions on boundary intervals

Analysis of Carleson measures for Dirichlet-Hardy spaces

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.