Lower bounds for the constants in the Bohnenblust-Hille inequality: the case of real scalars

TL;DR

This paper establishes new lower bounds for the constants in the Bohnenblust-Hille inequality for real scalars, complementing existing upper estimates and advancing understanding of the inequality's bounds.

Contribution

It provides the first non-trivial lower bounds for the constants in the Bohnenblust-Hille inequality for real scalars.

Findings

Established non-trivial lower bounds for the constants

Complemented existing upper bounds with new lower estimates

Enhanced understanding of the inequality's bounds

Abstract

The Bohnenblust-Hille inequality was obtained in 1931 and (in the case of real scalars) asserts that for every positive integer $N$ and every $m$-linear mapping $T:\ell_{\infty}^{N}\times...\times\ell_{\infty}^{N}\rightarrow \mathbb{R}$ one has (\sum\limits_{i_{1},...,i_{m}=1}^{N}|T(e_{i_{^{1}}},...,e_{i_{m}})|^{\frac{2m}{m+1}})^{\frac{m+1}{2m}}\leq C_{m}\VertT\Vert, for some positive constant $C_{m}$. Since then, several authors obtained upper estimates for the values of $C_{m}$. However, the novelty presented in this short note is that we provide lower (and non-trivial) bounds for $C_{m}$.