There exist multilinear Bohnenblust-Hille constants (Cn)n=1(infinity) with limn ->infinity(Cn+1-Cn)=0

Abstract

The n-linear Bohnenblust-Hille inequality asserts that there is a constant C-n is an element of [1, infinity) such that the l(2n/n+1)-norm of (U(e(i1), ..., e(in)))(i1, ...,in=1)(N) is bounded above by C-n times the supremum norm of U, for any n-linear form U :C-N x ... x C-N -> C and N is an element of N (the same holds for real scalars). We prove what we call Fundamental Lemma, which brings new information on the optimal constants, (K-n)(n=1)(infinity) for both real and complex scalars. For instance, Kn+1 - K-n < 0.87/n(0.473) for infinitely many n's. For complex scalars we give a formula (of surprisingly low growth), in which pi, e and the famous Euler-Mascheroni constant gamma appear: K-n < 1 (4/root pi (1 - e(gamma/2-1/2)) Sigma(n-1)(j=1) j(log2(e-gamma/2+1/2)-1)), for all(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 is an element of [2n/n+1, infinity). (C) 2012 Elsevier Inc. All rights reserved.