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oai:recercat.cat:2445/343642025-12-05T09:54:38Zcom_2072_1057col_2072_478917col_2072_478920

00925njm 22002777a 4500 dc Defant, Andreas author Frerick, Leonhard author Ortega Cerdà, Joaquim author Ounaïes, Myriam author Seip, Kristian author 2013-03-22T12:26:18Z 2013-03-22T12:26:18Z 2011 2013-03-22T12:26:18Z 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 $\sqrt{N}\exp\{(-1/\sqrt{2}+o(1))\sqrt{\log N\log\log N}\}$. Funcions de diverses variables complexes Funcions holomorfes Funcions de variables complexes Functions of several complex variables Holomorphic functions Functions of complex variables The Bonenblust-Hille inequality for homogeneous polynomials is hypercontractive