Analysis & PDE

  • Anal. PDE
  • Volume 7, Number 2 (2014), 513-527.

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

Daniel Carando, Andreas Defant, and Pablo Sevilla-Peris

Full-text: Open access

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<.

Article information

Source
Anal. PDE, Volume 7, Number 2 (2014), 513-527.

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

Permanent link to this document
https://projecteuclid.org/euclid.apde/1513731498

Digital Object Identifier
doi:10.2140/apde.2014.7.513

Mathematical Reviews number (MathSciNet)
MR3218818

Zentralblatt MATH identifier
1294.30008

Subjects
Primary: 30B50: Dirichlet series and other series expansions, exponential series [See also 11M41, 42-XX] 32A05: Power series, series of functions 46G20: Infinite-dimensional holomorphy [See also 32-XX, 46E50, 46T25, 58B12, 58C10]

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

Citation

Carando, Daniel; Defant, Andreas; Sevilla-Peris, Pablo. Bohr's absolute convergence problem for $\mathcal{H}_p$-Dirichlet series in Banach spaces. Anal. PDE 7 (2014), no. 2, 513--527. doi:10.2140/apde.2014.7.513. https://projecteuclid.org/euclid.apde/1513731498


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