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

2015/12/03 by Peloso, Marco M., Salvatori, Maura
#30C15 #30C40 #30H99 #46E22 #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1512.01452

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 \cal R. Such a measure has the form ω=ν⊗ m, where m is the Lebesgue measure on \mathbb R and ν is a regular Borel measure on [0,+∞). 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,+∞). We obtain a Paley-Wiener theorem for \cal M²_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≠2. 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.

Related

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