The asymptotic growth of the constants in the Bohnenblust-Hille inequality is optimal

Abstract

We provide (for both the real and complex settings) a family of constants, $% (C_{m})_{m\in \mathbb{N}}$, enjoying the Bohnenblust--Hille inequality and such that $\displaystyle\lim_{m\rightarrow \infty}\frac{C_{m}}{C_{m-1}}=1$, i.e., their asymptotic growth is the best possible. As a consequence, we also show that the optimal constants, $(K_{m})_{m\in \mathbb{N}}$, in the Bohnenblust--Hille inequality have the best possible asymptotic behavior. Besides its intrinsic mathematical interest and potential applications to different areas, the importance of this result also lies in the fact that all previous estimates and related results for the last 80 years (such as, for instance, the multilinear version of the famous Grothendieck Theorem for absolutely summing operators) always present constants $C_{m}$'s growing at an exponential rate of certain power of $m$.