The Bohnenblust-Hille inequality for homogeneous polynomials is hypercontractive

• 成果类型: Article
• 署名作者: Defant, Andreas; Frerick, Leonhard; Ortega-Cerda, Joaquim; Ounaies, Myriam; Seip, Kristian
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2011.174.1.13
• 发表日期: 2011
• 页码: 485-497
• 关键词: power-series theorem; dirichlet series; variables; SPACES; hardy

摘要

The Bohnenblust-Hille inequality says that the l(2m/m+1)-norm of the coefficients of an m-homogeneous polynomial P on C-n is bounded by parallel to P parallel to(infinity) times a constant independent of n, where parallel to.parallel to(infinity) denotes the supremum norm on the polydisc D-n. 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 D-n behaves asymptotically as root(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 root N exp {(-1/root 2+ o(1))root og N log log N} as N -> infinity.