On some spaces of holomorphic functions of exponential growth on a half-plane

Abstract

In this paper we study spaces of holomorphic functions on the right half-plane $\cal R$, that we denote by $\cal M^p_ω$, whose growth conditions are given in terms of a translation invariant measure $ω$ on the closed half-plane $\overline\cal R$. Such a measure has the form $ω=ν\otimes m$, where $m$ is the Lebesgue measure on $\mathbb R$ and $ν$ is a regular Borel measure on $[0,+\infty)$. We call these spaces generalized Hardy-Bergman spaces on the half-plane $\cal R$. We study in particular the case of $ν$ purely atomic, with point masses on an arithmetic progression on $[0,+\infty)$. We obtain a Paley-Wiener theorem for $\cal M^2_ω$, and consequentely the expression for its reproducing kernel. We study the growth of functions in such space and in particular show that $\cal M^p_ω$ contains functions of order 1. Moreover, we prove that the orthogonal projection from $L^p(\cal R,dω)$ into $\cal M^p_ω$ is unbounded for $p\neq2$. Furthermore, we compare the spaces $\cal M^p_ω$ with the classical Hardy and Bergman spaces, and some other Hardy-Bergman-type spaces introduced more recently.