Bohr’s absolute convergence problem for ℋp-Dirichlet series in Banach spaces

Carando, Daniel; Defant, Andreas; Sevilla-Peris, Pablo

doi:10.2140/apde.2014.7.513

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Bohr's absolute convergence problem for $\mathcal{H}_p$-Dirichlet series in Banach spaces

Daniel Carando, Andreas Defant and Pablo Sevilla-Peris

Vol. 7 (2014), No. 2, 513–527

DOI: 10.2140/apde.2014.7.513

Abstract

The Bohr–Bohnenblust–Hille theorem states that the width of the strip in the complex plane on which an ordinary Dirichlet series $\sum_{n} a_{n} n^{- s}$ converges uniformly but not absolutely is less than or equal to $\frac{1}{2}$ , and this estimate is optimal. Equivalently, the supremum of the absolute convergence abscissas of all Dirichlet series in the Hardy space $ℋ_{\infty}$ equals $\frac{1}{2}$ . By a surprising fact of Bayart the same result holds true if $ℋ_{\infty}$ is replaced by any Hardy space $ℋ_{p}$ , $1 \leq p < \infty$ , of Dirichlet series. For Dirichlet series with coefficients in a Banach space $X$ the maximal width of Bohr’s strips depend on the geometry of $X$ ; Defant, García, Maestre and Pérez-García proved that such maximal width equals $1 - 1 ∕ Cot X$ , where $Cot X$ denotes the maximal cotype of $X$ . Equivalently, the supremum over the absolute convergence abscissas of all Dirichlet series in the vector-valued Hardy space $ℋ_{\infty} (X)$ equals $1 - 1 ∕ Cot X$ . In this article we show that this result remains true if $ℋ_{\infty} (X)$ is replaced by the larger class $ℋ_{p} (X)$ , $1 \leq p < \infty$ .

Keywords

vector-valued Dirichlet series, vector-valued $H_p$ spaces, Banach spaces

Mathematical Subject Classification 2010

Primary: 30B50, 32A05, 46G20

Milestones

Received: 9 September 2013

Accepted: 2 January 2014

Published: 30 May 2014

Authors

Daniel Carando
Departamento de Matemática Universidad de Buenos Aires Ciudad Universitaria - Pabellón I C1428EGA Buenos Aires Argentina

Andreas Defant
Institut für Mathematik Universität Oldenburg D-26111 Oldenburg Germany

Pablo Sevilla-Peris
Instituto Universitario de Matemática Pura y Aplicada Universitat Politècnica de València 46022 València Spain

Vol. 7, No. 2, 2014