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2014 Bohr's absolute convergence problem for $\mathcal{H}_p$-Dirichlet series in Banach spaces
Daniel Carando, Andreas Defant, Pablo Sevilla-Peris
Anal. PDE 7(2): 513-527 (2014). 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 nanns converges uniformly but not absolutely is less than or equal to 12, and this estimate is optimal. Equivalently, the supremum of the absolute convergence abscissas of all Dirichlet series in the Hardy space equals 12. By a surprising fact of Bayart the same result holds true if is replaced by any Hardy space p, 1p<, 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 11CotX, where CotX denotes the maximal cotype of X. Equivalently, the supremum over the absolute convergence abscissas of all Dirichlet series in the vector-valued Hardy space (X) equals 11CotX. In this article we show that this result remains true if (X) is replaced by the larger class p(X), 1p<.

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Daniel Carando. Andreas Defant. Pablo Sevilla-Peris. "Bohr's absolute convergence problem for $\mathcal{H}_p$-Dirichlet series in Banach spaces." Anal. PDE 7 (2) 513 - 527, 2014. https://doi.org/10.2140/apde.2014.7.513

Information

Received: 9 September 2013; Accepted: 2 January 2014; Published: 2014
First available in Project Euclid: 20 December 2017

zbMATH: 1294.30008
MathSciNet: MR3218818
Digital Object Identifier: 10.2140/apde.2014.7.513

Subjects:
Primary: 30B50, 32A05, 46G20

Rights: Copyright © 2014 Mathematical Sciences Publishers

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