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arXiv · 1212.6672

A note on the hypercontractivity of the polynomial Bohnenblust--Hille inequality

Daniel Pellegrino
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Abstract

For $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$ and $m$ a positive integer, we remark that there is a constant $C$ so that, for all $r\in\lbrack1,\frac {2m}{m+1}],$ the supremum of the ratio between the $\ell_{r}$ norm of the coefficients of any nonzero $m$-homogeneous polynomial $P:\ell_{\infty}% ^{n}(\mathbb{K}) \rightarrow\mathbb{K}$ and its supremum norm is dominated by $C^{m}\cdot n^{(\frac{m}{r}-\frac{m+1}{2})}$ and, moreover, we prove that the exponent $\frac{m}{r}-\frac{m+1}{2}$ is optimal.

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Daniel Pellegrino. 2013-01-02. A note on the hypercontractivity of the polynomial Bohnenblust--Hille inequality. https://arxiv.org/abs/1212.6672

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