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

PUBLISHED BY mathematical sciences publishers nonprofit scientific publishing http://msp.org/ © 2014 Mathematical Sciences Publishers[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 < ∞Carando was partially supported by CONICET PIP 0624, PICT 2011-1456 and UBACyT 1-746. Defant and Sevilla-Peris were supported by MICINN project MTM2011-22417. Sevilla-Peris was partially supported by UPV-SP20120700.Carando, D.; Defant, A.; Sevilla Peris, P. (2014). BOHR'S ABSOLUTE CONVERGENCE PROBLEM FOR Hp-DIRICHLET SERIES IN BANACH SPACES. Analysis and PDE. 7(2):513-527. doi:10.2140/apde.2014.7.513S5135277