On unimodular multilinear forms with small norms on sequence spaces

Abstract

The Kahane–Salem–Zygmund inequality is a probabilistic result that guarantees the existence of special matrices with entries 1 and −1 generating unimodular m-linear forms Am,n:ℓp1n×⋯×ℓpmn⟶R (or C) with relatively small norms. The optimal asymptotic estimates for the smallest possible norms of Am,n when {p1,…,pm}⊂[2,∞] and when {p1,…,pm}⊂[1,2) are well-known and in this paper we obtain the optimal asymptotic estimates for the remaining case: {p1,…,pm} intercepts both [2,∞] and [1,2). In particular we prove that a conjecture posed by Albuquerque and Rezende is false and, using a special type of matrices that dates back to the works of Toeplitz, we also answer a problem posed by the same authors.