Let $p \ge 2$ . We improve the bound $\frac {\|f\|_{p}}{\|f\|_{2}} \le (p-1)^{s/2}$ for a polynomial $f$ of degree $s$ on the boolean cube $ \{0,1\}^{n}$ , which comes from hypercontractivity, replacing the right hand side of this inequality by an explicit bivariate function of $p$ and $s$ , which is smaller than $(p-1)^{s/2}$ for any $p > 2$ and $s > 0$ . We show the new bound to be tight, within a smaller order factor, for the Krawchouk polynomial of degree $s$ . This implies several nearly-extremal properties of Krawchouk polynomials and Hamming spheres (equivalently, Hamming balls). In particular, Krawchouk polynomials have (almost) the heaviest tails among all polynomials of the same degree and $\ell _{2}$ norm. ¹ The Hamming spheres have the following approximate edge-isoperimetric property: For all $1 \le s \le \frac {n}{2}$ , and for all even distances $0 \le i \le \frac {2s(n-s)}{n}$ , the Hamming sphere of radius $s$ contains, up to a multiplicative factor of $O(i)$ , as many pairs of points at distance $i$ as possible, among sets of the same size (there is a similar, but slightly weaker and somewhat more complicated claim for all distances). This also implies that Hamming spheres are (almost) stablest with respect to noise among sets of the same size. In coding theory terms this means that a Hamming sphere (equivalently a Hamming ball) has the maximal probability of undetected error, among all binary codes of the same rate. We also describe a family of hypercontractive inequalities for functions on $ \{0,1\}^{n}$ , which improve on the ‘usual’ “ $q \rightarrow 2$ ” inequality by taking into account the concentration of a function (expressed as the ratio between its $\ell _{r}$ norms), and which are nearly tight for characteristic functions of Hamming spheres. ¹

This has to be interpreted with some care.

中文翻译：

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多项式的矩比有界以及Krawchouk多项式和汉明球的极值性质

让 $ p \ ge 2 $ 。我们提高界限 $ \ frac {\ | f \ | __p}} {\ | f \ | _ {2}} \ le（p-1）^ {s / 2} $ 对于一个多项式 $ f $ 度 $ s $ 在布尔立方体上 $ \ {0,1 \} ^ {n} $ ，它来自于超收缩性，用的显式双变量函数代替了该不等式的右手边。 $ p $ 和 $ s $ ，小于 $（p-1）^ {s / 2} $ 对于任何 $ p> 2 $ 和 $ s> 0 $ 。我们显示，对于克劳乔克多项式 度 $ s $ 。这意味着Krawchouk多项式和汉明球体（等效地，汉明球）具有几个几乎极限的性质。尤其是，Krawchouk多项式在（几乎）相同阶数的所有多项式中具有（几乎）最重的尾部。 $ \ ell _ {2} $ 规范。 ^1个 海明球具有以下近似的边等距属性：对于所有 $ 1 \ le s \ le \ frac {n} {2} $ ，并且对于所有偶数距离 $ 0 \ le i \ le \ frac {2s（ns）} {n} $ ，半径的汉明球 $ s $ 包含，最多为 $ O（i）$ ，远处有多对点 $ i $ 尽可能在相同大小的集合中进行（对所有距离都具有相似但略微弱一点且更为复杂的要求）。这也意味着汉明球在大小相同的集合中相对于噪声而言（几乎）最稳定。用编码理论术语来说，这意味着在相同速率的所有二进制代码中，汉明球（等效为汉明球）具有最大的未被检测到的错误概率。我们还描述了一个关于函数的超收缩不等式族 $ \ {0,1 \} ^ {n} $ ，这是对“常规”的改进 $ q \ rightarrow 2 $ 通过考虑一个函数的集中度来表示不等式（表示为其之间的比率 $ \ ell _ {r} $ 规范），并且对于汉明球体的特征功能几乎是严格的。 ^1个

必须谨慎解释这一点。