Vol. 7, No. 2, 2014

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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

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 1 2, and this estimate is optimal. Equivalently, the supremum of the absolute convergence abscissas of all Dirichlet series in the Hardy space equals 1 2. By a surprising fact of Bayart the same result holds true if is replaced by any Hardy space p, 1 p < , 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 1CotX, 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 1 1CotX. In this article we show that this result remains true if (X) is replaced by the larger class p(X), 1 p < .

vector-valued Dirichlet series, vector-valued $H_p$ spaces, Banach spaces
Mathematical Subject Classification 2010
Primary: 30B50, 32A05, 46G20
Received: 9 September 2013
Accepted: 2 January 2014
Published: 30 May 2014
Daniel Carando
Departamento de Matemática
Universidad de Buenos Aires
Ciudad Universitaria - Pabellón I
C1428EGA Buenos Aires
Andreas Defant
Institut für Mathematik
Universität Oldenburg
D-26111 Oldenburg
Pablo Sevilla-Peris
Instituto Universitario de Matemática Pura y Aplicada
Universitat Politècnica de València
46022 València