Title:
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The Bonenblust-Hille inequality for homogeneous polynomials is hypercontractive
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Author:
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Defant, Andreas; Frerick, Leonhard; Ortega Cerdà, Joaquim; Ounaïes, Myriam; Seip, Kristian
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Other authors:
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Universitat de Barcelona |
Abstract:
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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 $\sqrt{N}\exp\{(-1/\sqrt{2}+o(1))\sqrt{\log N\log\log N}\}$. |
Subject(s):
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-Funcions de diverses variables complexes -Funcions holomorfes -Funcions de variables complexes -Functions of several complex variables -Holomorphic functions -Functions of complex variables |
Rights:
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(c) Annals of Mathematics, 2011
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Document type:
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Article Article - Published version |
Published by:
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Princeton University Press
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