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 $$(\sum\limits_{i_{1},...,i_{m}=1}^{N}|U(e_{i_{^{1}}},...,e_{i_{m}})| ^{\frac{2m}{m+1}}) ^{\frac{m+1}{2m}}\leq C_{m}| U|$$ for every positive integer $N$ and every $m$-linear mapping $U:\ell_{\infty}^{N}\times...\times\ell_{\infty}^{N}\rightarrow\mathbb{C}$, 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. Qu\'{e}ffelec 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 $2\leq m\leq 24$, 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.