New upper bounds for the constants in the Bohnenblust-Hille inequality

Abstract

A classical inequality due to Bohnenblust and Hille states that for every positive integer $m$ there is a constant $C_m>0$ so that $$(\sum _{i_1,...,i_m=1}^N|U(e_{i_{{}^1}},...,e_{i_m}){|}^{\frac{2m}{m+1}}{)}^{\frac{m+1}{2m}}\le C_m|U|$$ for every positive integer $N$ and every $m$-linear mapping $U:{\ell }_{\infty }^N\times ...\times {\ell }_{\infty }^N\to \mathbb{C}$, where $C_m=m^{\frac{m+1}{2m}}{2}^{\frac{m-1}{2}}.$ The value of $C_m$ was improved to $C_m=2^{\frac{m-1}{2}}$ by S. Kaijser and more recently H. Quéffelec and A. Defant and P. Sevilla-Peris remarked that $C_m=(\frac{2}{\sqrt{\pi }}{)}^{m-1}$ also works. The Bohnenblust--Hille inequality also holds for real Banach spaces with the constants $C_m=2^{\frac{m-1}{2}}$. In this note we show that a recent new proof of the Bohnenblust--Hille inequality (due to Defant, Popa and Schwarting) provides, in fact, quite better estimates for $C_m$ for all values of $m\in \mathbb{N}$. In particular, we will also show that, for real scalars, if $m$ is even with $2\le m\le 24$, then $$C_{\mathbb{R},m}=2^{1/2}{C}_{\mathbb{R},m/2}.$$ We will mainly work on a paper by Defant, Popa and Schwarting, giving some remarks about their work and explaining how to, numerically, improve the previously mentioned constants.

The present version improves the constants of the previous version