Abstract :
The search of sharp estimates for the constants in the Bohnenblust–Hille inequality, besides its challenging
nature, has quite important applications in different fields of mathematics and physics. For homogeneous
polynomials, it was recently shown that the Bohnenblust–Hille inequality (for complex scalars) is hypercontractive.
This result, interesting by itself, has found direct striking applications in the solution of
several important problems. For multilinear mappings, precise information on the asymptotic behavior of
the constants of the Bohnenblust–Hille inequality is of particular importance for applications in Quantum
Information Theory and multipartite Bell inequalities. In this paper, using elementary tools, we prove a quite
surprising result: the asymptotic growth of the constants in the multilinear Bohnenblust–Hille inequality is
optimal. Besides its intrinsic mathematical interest and potential applications to different areas, the mathematical
importance of this result also lies in the fact that all previous estimates and related results for the last
80 years (such as, for instance, the multilinear version of the famous Grothendieck theorem for absolutely
summing operators) always present constants Cm’s growing at an exponential rate of certain power of m.
© 2012 Elsevier Inc. All rights reserved.
