Mathematics – Complex Variables
Scientific paper
2009-04-23
Annals of Mathematics, vol 174, no1 , 2011, pp 485-497
Mathematics
Complex Variables
This paper supercedes partially the papers arXiv:0903.1455 and arXiv:0903.3395 and obtains new applications
Scientific paper
10.4007/annals.2011.174.1.13
The Bohnenblust--Hille inequality says that the $\ell^{\frac{2m}{m+1}}$-norm of the coefficients of an $m$-homogeneous polynomial $P$ on $\C^n$ is bounded by $\| P\|_\infty$ times a constant independent of $n$, where $\|\cdot \|_\infty$ 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 $\sqrt{(\log n)/n}$ 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 $\sqrt{N}\exp\{(-1/\sqrt{2}+o(1))\sqrt{\log N\log\log N}\}$ as $N\to \infty$.
Defant Andreas
Frerick Leonhard
Ortega-Cerdà Joaquim
Ounaies Myriam
Seip Kristian
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