Some improvements on the constants for the real Bohnenblust-Hille inequality

Abstract: A classical inequality due to Bohnenblust and Hille states that for every $N \in \mathbb{N}$ and every $m$-linear mapping $U:\ell_{\infty}^{N}\times...\times\ell_{\infty}^{N}\rightarrow\mathbb{C}$ we have [(\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|] where $C_{m}=2^{\frac{m-1}{2}}$. The result is also true for real Banach spaces. In this note we show that an adequate use of a recent new proof of Bohnenblust-Hille inequality, due to Defant, Popa and Schwarting, combined with the optimal constants of Khinchine's inequality (due to Haagerup) provides quite better estimates for the constants involved in the real Bohnenblust-Hille inequality. For instance, for $2\leq m\leq 14,$ we show that the constants $C_{m}=2^\frac{m-1}{2}$ can be replaced by $2^{\frac{m^{2}+6m-8}{8m}}$ if $m$ is even and by $2^{\frac{m^{2}+6m-7}{8m}}$ if $m$ is odd, which substantially improve the known values of $C_{m}$. We also show that the new constants present a better asymptotic behavior.