Estimates for the asymptotic behavior of the constants in the Bohnenblust--Hille inequality

Abstract

A classical inequality due to H.F. Bohnenblust and E. Hille states that for every positive integer $n$ there is a constant $C_n>0$ so that $$(\sum _{i_1,...,i_n=1}^N|U(e_{i_{{}^1}},...,e_{i_n}){|}^{\frac{2n}{n+1}}{)}^{\frac{n+1}{2n}}\le C_n||U||$$ for every positive integer $N$ and every $n$-linear mapping $U:{\ell }_{\infty }^N\times ...\times {\ell }_{\infty }^N\to \mathbb{C}$. The original estimates for those constants from Bohnenblust and Hille are $$C_n=n^{\frac{n+1}{2n}}{2}^{\frac{n-1}{2}}.$$ In this note we present explicit formulae for quite better constants, and calculate the asymptotic behavior of these estimates, completing recent results of the second and third authors. For example, we show that, if $C_{\mathbb{R},n}$ and $C_{\mathbb{C},n}$ denote (respectively) these estimates for the real and complex Bohnenblust--Hille inequality then, for every even positive integer $n$, $$\frac{C_{\mathbb{R},n}}{\sqrt{\pi }}=\frac{C_{\mathbb{C},n}}{\sqrt{2}}=2^{\frac{n+2}{8}}\cdot r_n$$ for a certain sequence $\{r_n\}$ which we estimate numerically to belong to the interval $(1,3/2)$ (the case $n$ odd is similar). Simultaneously, assuming that $\{r_n\}$ is in fact convergent, we also conclude that $$\underset{n\to \infty }{\lim }\frac{C_{\mathbb{R},n}}{C_{\mathbb{R},n-1}}=\underset{n\to \infty }{\lim }\frac{C_{\mathbb{C},n}}{C_{\mathbb{C},n-1}}=2^{1/8}.$$

7 pages

References in corpus (2)

• The Bohnenblust--Hille inequality for homogeneous polynomials is hypercontractive
• The Grothendieck constant is strictly smaller than Krivine's bound