There exist multilinear Bohnenblust-Hille constants $(C_{n})_{n=1}^{\infty}$ with $\displaystyle \lim_{n\rightarrow \infty}(C_{n+1}-C_{n}) =0.$

Abstract

The $n$-linear Bohnenblust-Hille inequality asserts that there is a constant $C_{n}\in\lbrack1,\infty)$ such that the $\ell_{\frac{2n}{n+1}}$-norm of $(U(e_{i_{^{1}}},...,e_{i_{n}}))_{i_{1},...i_{n}=1}^{N}$is bounded above by $C_{n}$ times the supremum norm of $U,$ regardless of the $n$-linear form $U:\mathbb{C}^{N}\times...\times\mathbb{C}^{N}% \rightarrow\mathbb{C}$ and the positive integer $N$ (the same holds for real scalars). The power $2n/(n+1)$ is sharp but the values and asymptotic behavior of the optimal constants remain a mystery. The first estimates for these constants had exponential growth. Very recently, a new panorama emerged and the importance, for many applications, of the knowledge of the optimal constants (denoted by $(K_{n})_{n=1}^{\infty}$) was stressed. The title of this paper is part of our Fundamental Lemma, one of the novelties presented here. It brings surprising new (and precise) information on the optimal constants (for both real and complex scalars). For instance, [K_{n+1}-K_{n}<\frac{0.87}{n^{0.473}}] for infinitely many $n$'s. In the case of complex scalars we present a curious formula, where $\pi,e$ and the famous Euler--Mascheroni constant $\gamma$ appear together: [K_{n}<1+(\frac{4}{\sqrt{\pi}}(1-e^{\gamma/2-1/2}) {\sum\limits_{j=1}^{n-1}}j^{^{\log_{2}(e^{-\gamma/2+1/2}) -1}%})] for all $n\geq2$. Numerically, the above formula shows a surprising low growth, [K_{n}<1.41(n-1)^{0.305}-0.04] for every integer $n \geq2$. We also provide a brief discussion on the interplay between the Kahane-Salem-Zygmund and the Bohnenblust-Hille (polynomial and multilinear) inequalities.