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

DANIEL CARANDO

Congreso; Mathematical Congress of the Americas; 2013

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 $1- 1/Cot(X)$, where $Cot(X)$ denotes the optimal 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 $1- 1/Cot(X)$.In this talk we will revisit these facts and present our contribution: the last result remains true if  $mathcal{H}_infty(X)$ is replaced by the larger class $mathcal{H}_p(X)$, $1 le p < infty$.