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

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