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The Bohnenblust-Hille inequality for homogeneous polynomials is hypercontractive. (English) Zbl 1235.32001
The main goal of this nice and important paper is to obtain that the Bohnenblust-Hille inequality is hypercontractive. The Bohnenblust-Hille inequality says that the \(\ell^{\frac{2m}{m+1}}\)-norm of the coefficients of an \(m\)-homogeneous polynomial on \(\mathbb{C}^n\) is bounded above by the supremum norm of this polynomial times a constant which does not depend on \(n\). The authors show in fact that this constant can be taken to be \(C^m\) for some \(C>1\). This result considerably improves the known bounds, see [L. A. Harris, Analyse fonct. Appl., C. r. Colloq. d’Analyse, Rio de Janeiro 1972, 145–163 (1975; Zbl 0315.46040); H. Queffélec, J. Anal. 3, 43–60 (1995; Zbl 0881.11068)] and leads to some interesting consequences. For instance, the authors obtain an asymptotically optimal estimate for the \(n\)-dimensional Bohr radius, which was introduced and studied by H. P. Boas and D. Khavinson [Proc. Am. Math. Soc. 125, No. 10, 2975–2979 (1997; Zbl 0888.32001)], as well as a refined version of a striking theorem of S. V. Konyagin and H. Queffélec [Real Anal. Exch. 27, No. 1, 155–175 (2002; Zbl 1026.42011)] on Dirichlet polynomials that was recently sharpened by R. de la Bretèche [Acta Arith. 134, No. 2, 141–148 (2008; Zbl 1302.11068)]. In the same spirit of the latter, the authors give the following stronger version of a result of R. Balasubramanian, B. Calado and H. Queffélec [Stud. Math. 175, No. 3, 285–304 (2006; Zbl 1110.30001)]: they prove that \(1/\sqrt{2}\) is the supremum of the set of real numbers \(c\) such that
\[ \sum_{n\geq 1}| a_n| n^{-\frac{1}{2}}\text{exp}\Big\{c\sqrt{\log n\log \log n}\Big\}<\infty \]
for every ordinary Dirichlet series \(f(s)=\sum_{n\geq 1}a_nn^{-s}\) for which \(\sup\{| f(\sigma+it)| :\;\sigma>0\}<\infty.\)

MSC:
32A22 Nevanlinna theory; growth estimates; other inequalities of several complex variables
30B50 Dirichlet series, exponential series and other series in one complex variable
32A05 Power series, series of functions of several complex variables
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