Defant, AndreasFrerick, LeonhardOrtega Cerdà, JoaquimOunaïes, MyriamSeip, Kristian2013-03-222013-03-2220110003-486Xhttps://hdl.handle.net/2445/34364The Bohnenblust-Hille inequality says that the $\ell^{\frac{2m}{m+1}}$ -norm of the coefficients of an $m$-homogeneous polynomial $P$ on $\Bbb{C}^n$ is bounded by $\| P \|_\infty$ times a constant independent of $n$, where $\|\cdot \|_\infty$ denotes the supremum norm on the polydisc $\mathbb{D}^n$. 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 $\mathbb{D}^n$ behaves asymptotically as $\sqrt{(\log n)/n}$ modulo a factor bounded away from 0 and infinity, and the Sidon constant for the set of frequencies $\bigl\{ \log n: n \text{a positive integer} \le N\bigr\}$ is $\sqrt{N}\exp\{(-1/\sqrt{2}+o(1))\sqrt{\log N\log\log N}\}$.13 p.application/pdfeng(c) Annals of Mathematics, 2011Funcions de diverses variables complexesFuncions holomorfesFuncions de variables complexesFunctions of several complex variablesHolomorphic functionsFunctions of complex variablesinfo:eu-repo/semantics/article5831992013-03-22info:eu-repo/semantics/openAccess