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 $\frac{1}{2}$, and this estimate is optimal. Equivalently, the supremum of the absolute convergence abscissas of all Dirichlet series in the Hardy space ${\mathcal{\mathscr{H}}}_{\infty }$ equals $\frac{1}{2}$. By a surprising fact of Bayart the same result holds true if ${\mathcal{\mathscr{H}}}_{\infty }$ is replaced by any Hardy space ${\mathcal{\mathscr{H}}}_p$, $1\le p<\infty$, 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 $1-1∕CotX$, 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 ${\mathcal{\mathscr{H}}}_{\infty }\left(X\right)$ equals $1-1∕CotX$. In this article we show that this result remains true if ${\mathcal{\mathscr{H}}}_{\infty }\left(X\right)$ is replaced by the larger class ${\mathcal{\mathscr{H}}}_p\left(X\right)$, $1\le p<\infty$.