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

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 1/2, and this estimate is optimal. Equivalently, the supremum of the absolute convergence abscissas of all Dirichlet series in the Hardy space ${\mathcal{H}}_{\infty }$ equals 1/2. By a surprising fact of Bayart the same result holds true if ${\mathcal{H}}_{\infty }$ is replaced by any Hardy space ${\mathcal{H}}_p$, $1\le p<\infty$, 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 equal $1- 1/\ct(X)$, where $\ct(X)$ denotes the maximal cotype of $X$. Equivalently, the supremum over the absolute convergence abscissas of all Dirichlet series in the vector-valued Hardy space ${\mathcal{H}}_{\infty }(X)$ equals $1- 1/\ct(X)$. In this article we show that this result remains true if ${\mathcal{H}}_{\infty }(X)$ is replaced by the larger class ${\mathcal{H}}_p(X)$, $1\le p<\infty$.

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