#### 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
##### 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 ${\mathsc{ℋ}}_{\infty }$ equals $\frac{1}{2}$. By a surprising fact of Bayart the same result holds true if ${\mathsc{ℋ}}_{\infty }$ is replaced by any Hardy space ${\mathsc{ℋ}}_{p}$, $1\le p<\infty$, 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 $1-1∕\phantom{\rule{0.3em}{0ex}}CotX$, 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 ${\mathsc{ℋ}}_{\infty }\left(X\right)$ equals $1-1∕CotX$. In this article we show that this result remains true if ${\mathsc{ℋ}}_{\infty }\left(X\right)$ is replaced by the larger class ${\mathsc{ℋ}}_{p}\left(X\right)$, $1\le p<\infty$.

##### Keywords
vector-valued Dirichlet series, vector-valued $H_p$ spaces, Banach spaces
##### Mathematical Subject Classification 2010
Primary: 30B50, 32A05, 46G20