Bohr's absolute convergence problem for Hp-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 ∑nann-s converges uniformly but not absolutely is less than or equal to 12, and this estimate is optimal. Equivalently, the supremum of the absolute convergence abscissas of all Dirichlet series in the Hardy space H1 equals 1/2. By a surprising fact of Bayart the same result holds true if H1 is replaced by any Hardy space H∞, 1 ≤ p <∞, 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=Cot X, where Cot 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 H∞.(X) equals 1-1/Cot X. In this article we show that this result remains true if H∞(X) is replaced by the larger class Hp.(X), 1 ≤ p < ∞.