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

AbstractThe n-linear Bohnenblust–Hille inequality asserts that there is a constant Cn∈[1,∞) such that the ℓ2nn+1-norm of (U(ei1,…,ein))i1,…,in=1N 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<0.87n0.473 for infinitely many nʼs. For complex scalars we give a formula (of surprisingly low growth), in which π,e and the famous Euler–Mascheroni constant γ appear:Kn