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ABSTRACT. The Bohnenblust-Hille inequality says that the ℓ 2m m+1 -norm of the coefficients of an m-homogeneous polynomial P on C n is bounded by P ∞ times a constant independent of n, where · ∞ denotes the supremum norm on the polydisc D n . The main result of this paper is that this inequality is hypercontractive, i.e., the constant can be taken to be C m for some C > 1. Combining this improved version of the Bohnenblust-Hille inequality with other results, we obtain the following: The Bohr radius for the polydisc D n behaves asymptotically as (log n)/n modulo a factor bounded away from 0 and infinity, and the Sidon constant for the set of frequencies log n : n a positive integer ≤ N is √ N exp{(−1/ √ 2 + o(1)) √ log N log log N } as N → ∞.

We invent the new notion of coordinatewise multiple summing operators in Banach spaces, and use it to study various vector valued extensions of the well-know Bohnenblust-Hille inequality (which originally extended Littlewood's 4/3-inequality). Our results have application on the summability of monomial coefficients of m-homogeneous polynomials P : ∞ → p , as well as for the convergence theory of products of vector valued Dirichlet series.

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Bohr showed that the width of the strip (in the complex plane) on which a given Dirichlet series a n /n s , s ∈ C, converges uniformly but not absolutely, is at most 1/2, and Bohnenblust-Hille that this bound in general is optimal. We prove that for a given infinite dimensional Banach space Y the width of Bohr's strip for a Dirichlet series with coefficients a n in Y is bounded by 1 − 1/ Cot(Y ), where Cot(Y ) denotes the optimal cotype of Y . This estimate even turns out to be optimal, and hence leads to a new characterization of cotype in terms of vector valued Dirichlet series. Mathematics Subject Classification (2000)Primary 32A05; Secondary 46B07 · 46B09 · 46G20 The first, second and third authors were supported by MEC and FEDER Project MTM2005-08210. A. Defant

We unify Littlewood's classical 4/3-inequality (a forerunner of Grothendieck's inequality) together with its m-linear extension due to Bohnenblust and Hille (which originally settled Bohr's absolute convergence problem for Dirichlet series) with a scale of inequalties of Bennett and Carl in p -spaces (which are of fundamental importance in the theory of eigenvalue distribution of power compact operators). As an application we give estimates for the monomial coefficients of homogeneous p -valued polynomials on c 0 .

A result of Helson on general Dirichlet series a n e −λns states that, whenever (a n ) is 2-summable and λ = (λ n ) satisfies a certain condition introduced by Bohr, then for almost all homomorphism ω : (R, +) → T the Dirichlet series a n ω(λ n )e −λns converges on the open right half plane [Re > 0]. For ordinary Dirichlet series a n n −s Hedenmalm and Saksman related this result with the famous Carleson-Hunt theorem on pointwise convergence of Fourier series, and Bayart extended it within his theory of Hardy spaces H p of such series. The aim here is to prove variants of Helson's theorem within our recent theory of Hardy spaces H p (λ), 1 ≤ p ≤ ∞, of general Dirichlet series. To be more precise, in the reflexive case 1 < p < ∞ we extend Helson's result to Dirichlet series in H p (λ) without any further condition on the frequency λ, and in the non-reflexive case p = 1 to the wider class of frequencies satisfying the so-called Landau condition (more general than Bohr's condition). In both cases we add relevant maximal inequalities. Finally, we give several applications to the structure theory of Hardy spaces of general Dirichlet series.