Hadamard Rigidity and Sharp Stability in the Completely Bounded Bohnenblust--Hille Inequality
Abstract
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≥2, we also establish sharp structural stability. A form with completely bounded norm one and deficit ε lies within Cd√ε, in the critical coefficient ℓ2d/(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.
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BibTeX
@misc{nunezalarcon2026hadamard,
title = {{Hadamard Rigidity and Sharp Stability in the Completely Bounded Bohnenblust--Hille Inequality}},
author = {Daniel Núñez-Alarcón and Daniel M. Pellegrino and Eduardo V. Teixeira},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16329},
eprint = {2609.16329},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16329},
}
RIS
TY - UNPB
TI - Hadamard Rigidity and Sharp Stability in the Completely Bounded Bohnenblust--Hille Inequality
AU - Daniel Núñez-Alarcón
AU - Daniel M. Pellegrino
AU - Eduardo V. Teixeira
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.16329
UR - https://arxiv.org/abs/2609.16329
AB - 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≥2, we also establish sharp structural stability. A form with completely bounded norm one and deficit ε lies within C_d√ε, in the critical coefficient ℓ_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.
ER -