The Bohnenblust--Hille inequality for homogeneous polynomials is hypercontractive

Abstract

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 $\Vert P{\Vert }_{\infty }$ times a constant independent of $n$, where $\Vert \cdot {\Vert }_{\infty }$ 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 $\{\log n:n\text{a positive integer}\le N\}$ is $\sqrt{N}\exp \{(-1/\sqrt{2}+o(1))\sqrt{\log N\log \log N}\}$ as $N\to \infty$.

This paper supercedes partially the papers arXiv:0903.1455 and arXiv:0903.3395 and obtains new applications

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