Lower bounds for the constants in the Bohnenblust–Hille inequality: The case of real scalars

Abstract:

The Bohnenblust–Hille inequality was obtained in 1931 and (in the case of real scalars) asserts that for every positive integer $m$ there is a constant $C_{m}$ so that \begin{equation*} \left ( \sum \limits _{i_{1},...,i_{m}=1}^{N}\left \vert T(e_{i_{^{1}}},...,e_{i_{m}})\right \vert ^{\frac {2m}{m+1}}\right ) ^{\frac {m+1}{2m}}\leq C_{m}\left \Vert T\right \Vert \end{equation*} for all positive integers $N$ and every $m$-linear mapping $T:\ell _{\infty }^{N}\times \cdots \times \ell _{\infty }^{N}\rightarrow \mathbb {R}$. Since then, several authors have obtained upper estimates for the values of $C_{m}$. However, the novelty presented in this short note is that we provide lower (and non-trivial) bounds for $C_{m}$.