The Bohnenblust-Hille inequality for homogeneous polynomials is hypercontractive

Abstract

The Bohnenblust-Hille inequality says that the ℓ2m/m+1 -norm of the coefcients of an m-homogeneous polynomial P on Cn is bounded by kPk1 times a constant independent of n, where || · ||∞denotes the supremum norm on the polydisc Dn. The main result of this paper is that this inequality is hypercontractive, i.e., the constant can be taken to be Cm for some C > 1. Combining this improved version of the Bohnenblust-Hille inequality with other results, we obtain the following: The Bohr radius for the polydisc Dn behaves asymptotically as n modulo a factor bounded away from 0 and infinity, and the Sidon constant for the set of frequencies {log n: n a positive integer ≤ N} as N → ∞.