Abstract :
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.
