完全有界Bohnenblust--Hille不等式中的Hadamard刚性及尖锐稳定性
Hadamard Rigidity and Sharp Stability in the Completely Bounded Bohnenblust--Hille Inequality
- Universidad Nacional de Colombia（哥伦比亚国立大学）
- Universidade Federal da Paraíba（帕拉伊巴联邦大学）
- Oklahoma State University（俄克拉荷马州立大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该论文对完全有界Bohnenblust--Hille不等式进行了等式分类，证明非零等式情形为Hadamard链的标量倍数，并建立了亏损为ε时的尖锐结构稳定性，常数与维数无关且指数1/2最优。
AI中文摘要：
Arunachalam、Dutt、Escudero Gutiérrez和Palazuelos提出的完全有界Bohnenblust--Hille不等式以最优常数1控制系数可和性，且该常数与周围维数无关。我们在实域和复域上对所有非零等式情形进行了分类：它们恰好是支撑在等边笛卡尔盒子上的Hadamard链的标量倍数。因此，等式同时决定了系数张量的支撑和分解。对于每个固定次数$d\ge2$，我们还建立了尖锐的结构稳定性。一个完全有界范数为1且亏损为$\varepsilon$的型，在临界系数$\ell_{2d/(d+1)}$范数下，位于同一盒子上的平坦幺模路径和归一化酉链的$C_d\sqrt\varepsilon$范围内。相应的幺模和缩放酉边满足匹配的归一化Frobenius估计。在零亏损时，两个逼近都与原始Hadamard链重合。所有常数均与周围维数无关，且指数$1/2$在任一域上对两种逼近都是最优的。
英文摘要：
The completely bounded Bohnenblust--Hille inequality of Arunachalam, Dutt, Escudero Gutiérrez and Palazuelos controls coefficient summability with optimal constant one, independently of the ambient dimension. We classify all nonzero equality cases over both the real and complex fields: they are precisely scalar multiples of Hadamard chains supported on equal-sided Cartesian boxes. Thus equality determines both the support and the factorization of the coefficient tensor. For each fixed degree $d\ge2$, we also establish sharp structural stability. A form with completely bounded norm one and deficit $\varepsilon$ lies within $C_d\sqrt\varepsilon$, in the critical coefficient $\ell_{2d/(d+1)}$ norm, of both a flat unimodular path and a normalized unitary chain on the same box. The corresponding unimodular and scaled unitary edges satisfy a matching normalized Frobenius estimate. At zero deficit both approximants coincide with the original Hadamard chain. All constants are independent of the ambient dimension, and the exponent $1/2$ is optimal for both approximations over either field.
