关于齐次多项式的Bohnenblust-Hille不等式是超压缩的
The Bohnenblust--Hille inequality for homogeneous polynomials is hypercontractive
摘要：
Bohnenblust-Hille不等式说$\C^n$上的$m$ -齐次多项式$P$的系数的$\ell^{\frac{2m}{m+1}}$ -范数由$\| P\|_\infty$乘以一个与$n$无关的常数有界，其中$\|\cdot \|_\infty$表示多圆盘$\D^n$上的上确界范数。本文的主要结果是这个不等式是超压缩的，即，对于某些$C>1$，该常数可以取为$C^m$。结合这个改进的Bohnenblust-Hille不等式和其它结果，我们得到：多圆盘$\D^n$的玻尔半径渐近地表现为$\sqrt{(\log n)/n}$模一个从0到无穷远的因子，频率集$\bigl\{\log n: n \text{a positive integer} \le N\bigr\}$的西顿常数为$\sqrt{N}\exp\{(-1/\sqrt{2}+o(1))\sqrt{\log N\log\log N}\}$ as $N\to \infty$ .
英文摘要：
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 $\| P\|_\infty$ times a constant independent of $n$, where $\|\cdot \|_\infty$ 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 $\bigl\{\log n: n \text{a positive integer} \le N\bigr\}$ is $\sqrt{N}\exp\{(-1/\sqrt{2}+o(1))\sqrt{\log N\log\log N}\}$ as $N\to \infty$.
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