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

TL;DR

This paper extends Bohr's absolute convergence problem for Dirichlet series from Hardy spaces inity to rom p, showing the maximal width of convergence strips depends on the Banach space's geometry, specifically its cotype.

Contribution

It proves that the maximal width of Bohr's strips in vector-valued Hardy spaces rom p remains determined by the Banach space's cotype, generalizing previous results for inity.

Findings

Maximal width of Bohr's strips depends on Banach space cotype.

The result holds for Hardy spaces rom p, 1 ss p < inity.

Extension of classical Dirichlet series convergence results to vector-valued 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, Garc\'ia, Maestre and P\'erez-Garc\'ia 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 $\mathcal{H}_p(X)$, $1 \le p < \infty$.