There exist multilinear Bohnenblust–Hille constants (Cn) ∞ n=1 with limn→∞(Cn+1 − Cn) = 0

The n-linear Bohnenblust–Hille inequality asserts that there is a constant Cn ∈ [1,∞) such that the 2n n+1 -norm of (U(ei1, . . . , ein ))N i1,...,in=1 is bounded above by Cn times the supremum norm of U, for any n-linear form U : CN ×· · ·×CN →C and N ∈ N (the same holds for real scalars). We prove what we call Fundamental Lemma, which brings new information on the optimal constants, (Kn) ∞ n=1, for both real and complex scalars. For instance, Kn+1 −Kn < 0.87 n0.473 for infinitely many n’s. For complex scalars we give a formula (of surprisingly low growth), in which π, e and the famous Euler–Mascheroni constant γ appear:Kn < 1 + 4 √ π 1 −eγ/2−1/2 n −1 j=1 j log2(e −γ/2+1/2)−1 , ∀n 2. We study the interplay between the Kahane–Salem–Zygmund and the Bohnenblust–Hille (polynomial and multilinear) inequalities and provide estimates for Bohnenblust–Hille-type inequality constants for any exponent q ∈ [ 2n n+1 ,∞). © 2012 Elsevier Inc. All rights reserved.