论文标题
在序列空间上具有较小规范的单型多线性形式上
On unimodular multilinear forms with small norms on sequence spaces
论文作者
论文摘要
Kahane-salem-- Zygmund不平等是一个概率结果，可确保存在带有条目的特殊矩阵$ 1 $和-1 $生成un-dimodular $ m $ m $ - linear forms $ a_ {m，n}：\ ell _ elly_ \ cdots \ times \ ell_ {p_ {m}}^{n} \ longrightArrow \ Mathbb {r} $（或$ \ Mathbb {C} $）具有相对较小的规范。 $ \ \ left \ left \ {p_ {1}，...，p_ {m} \ right \} \ subset \ lbrack2，\ lbrack2，\ infty] $，当$ \ {p_ {1}，... \ subset \ lbrack1,2）$是众所周知的，在本文中，我们获得了剩余情况的最佳渐近估计值：$ \ left \ {p_ {1}，...，p_ {m} \ right \} $ intercepts $ intercepts $ [2，\ infty] $ [2，\ infty] $和$ [1,2 $ [1,2）$。特别是我们证明，阿尔伯克基（Albuquerque）和雷津德（Rezende）提出的猜想是错误的，并且使用一种特殊类型的矩阵，可追溯到toeplitz的作品，我们还回答了同一作者提出的问题。
The Kahane--Salem--Zygmund inequality is a probabilistic result that guarantees the existence of special matrices with entries $1$ and $-1$ generating unimodular $m$-linear forms $A_{m,n}:\ell_{p_{1}}^{n}\times \cdots\times\ell_{p_{m}}^{n}\longrightarrow\mathbb{R}$ (or $\mathbb{C}$) with relatively small norms. The optimal asymptotic estimates for the smallest possible norms of $A_{m,n}$ when $\left\{ p_{1},...,p_{m}\right\} \subset\lbrack2,\infty]$ and when $\left\{ p_{1},...,p_{m}\right\} \subset\lbrack1,2)$ are well-known and in this paper we obtain the optimal asymptotic estimates for the remaining case: $\left\{ p_{1},...,p_{m}\right\} $ intercepts both $[2,\infty]$ and $[1,2)$. In particular we prove that a conjecture posed by Albuquerque and Rezende is false and, using a special type of matrices that dates back to the works of Toeplitz, we also answer a problem posed by the same authors.
