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BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access April 25, 2016

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

Marco M. Peloso and Maura Salvatori
From the journal Concrete Operators

Abstract

In this paper we study spaces of holomorphic functions on the right half-plane R, that we denote by Mpω, whose growth conditions are given in terms of a translation invariant measure ω on the closed half-plane R. Such a measure has the form ω = ν ⊗ m, where m is the Lebesgue measure on R and ν is a regular Borel measure on Œ[0, +∞). We call these spaces generalized Hardy–Bergman spaces on the half-plane R. We study in particular the case of ν purely atomic, with point masses on an arithmetic progression on Œ[0, +∞). We obtain a Paley–Wiener theorem for M2ω, and consequentely the expression for its reproducing kernel. We study the growth of functions in such space and in particular show that Mpω contains functions of order 1. Moreover, we prove that the orthogonal projection from Lp(R,dω) into Mpω is unbounded for p ≠ 2. Furthermore, we compare the spaces Mpω with the classical Hardy and Bergman spaces, and some other Hardy– Bergman-type spaces introduced more recently.

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Received: 2015-11-9
Accepted: 2016-4-5
Published Online: 2016-4-25

© 2016 Peloso and Salvatori

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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