BOHR'S ABSOLUTE CONVERGENCE PROBLEM FOR Hp-DIRICHLET SERIES IN BANACH SPACES

Abstract

[EN] The Bohr–Bohnenblust–Hille theorem states that the width of the strip in the complex plane on which an ordinary Dirichlet series P n ann −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 H∞ equals 1 2 . By a surprising fact of Bayart the same result holds true if H∞ is replaced by any Hardy space Hp, 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 < ∞