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 $\Bbb{C}^n$ 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 $\sqrt{N}\exp\{(-1/\sqrt{2}+o(1))\sqrt{\log N\log\log N}\}$ as $N\to \infty$.

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Authors

Andreas Defant

Institut für Mathematik
Carl von Ossietzky Universität Oldenburg
26111 Oldenburg
Germany

Leonhard Frerick

FB IV - Mathematik
Universität Trier
54286 Trier
Germany

Joaquim Ortega-Cerdà

Facultat de Matemàtiques
Universitat de Barcelona,
08007Barcelona
Spain

Myriam Ounaïes

Université de Strasbourg
IRMA
7 Rue René Descartes
67084 Strasbourg Cedex
France

Kristian Seip

Department of Mathematical Sciences
Norwegian University of Science and Technology
NO-7491 Trondheim
Norway