|
|||||
 |
|
|
|
|
|
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
|
Abstract |
|
The Bohr–Bohnenblust–Hille theorem states that the width of the strip in the complex plane on which an ordinary Dirichlet series converges uniformly but not absolutely is less than or equal to , and this estimate is optimal. Equivalently, the supremum of the absolute convergence abscissas of all Dirichlet series in the Hardy space equals . By a surprising fact of Bayart the same result holds true if is replaced by any Hardy space , , of Dirichlet series. For Dirichlet series with coefficients in a Banach space the maximal width of Bohr’s strips depend on the geometry of ; Defant, García, Maestre and Pérez-García proved that such maximal width equals , where denotes the maximal cotype of . Equivalently, the supremum over the absolute convergence abscissas of all Dirichlet series in the vector-valued Hardy space equals . In this article we show that this result remains true if is replaced by the larger class , . |
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 |
|
|
