The Bohnenblust--Hille inequality for homogeneous polynomials is hypercontractive

2009/04/23 by Andreas Defant, Leonhard Frerick, Joaquim Ortega-Cerdà +2
Mathematics · #math.CV #math.FA #msc:32A05 #msc:43A46

paper · pdf · doi:10.4007/annals.2011.174.1.13

published as Annals of Mathematics, vol 174, no1 , 2011, pp 485-497 · This paper supercedes partially the papers arXiv:0903.1455 and arXiv:0903.3395 and obtains new applications

arxiv created 2009/04/23 · arxiv updated 2011/10/06

Abstract

The Bohnenblust--Hille inequality says that the ℓ^(2m)/(m+1)-norm of the coefficients of an m-homogeneous polynomial P on \Cⁿ is bounded by ‖ P‖_∞ 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 C^m 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 √((log n)/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\ is √(N)exp\(-1/√(2)+o(1))√(log Nloglog N)\ as N→ ∞.

Citations

• The Sidon constant for homogeneous polynomials
• Hypercontractivity of the Bohnenblust-Hille inequality for polynomials and multidimensional Bohr radii

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