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 Cm so that

((N)Sigma(i1 , . . . , im=1)vertical bar T(e(i1) (,...,) e(im))vertical bar(2m/m+1))(m+1/2) <= C-m parallel to T parallel to

for all positive integers N and every m-linear mapping T : l(infinity)(N) x...x l(infinity)(N) -> 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.