The asymptotic growth of the constants in the Bohnenblust–Hille inequality is optimal

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 Cmʼs growing at an exponential rate of certain power of m.