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 $N$ and every $m$-linear mapping $T:{\ell }_{\infty }^N\times ...\times {\ell }_{\infty }^N\to \mathbb{R}$ one has (\sum\limits_{i_{1},...,i_{m}=1}^{N}|T(e_{i_{^{1}}},...,e_{i_{m}})|^{\frac{2m}{m+1}})^{\frac{m+1}{2m}}\leq C_{m}\VertT\Vert, for some positive constant $C_m$. Since then, several authors 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 $C_m$.

References in corpus (2)

• The Bohnenblust--Hille inequality for homogeneous polynomials is hypercontractive
• The asymptotic growth of the constants in the Bohnenblust-Hille inequality is optimal

Cited by in corpus (1)

• A geometric technique to generate lower estimates for the constants in the Bohnenblust--Hille inequalities