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

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Diniz, D. and Muñoz-Fernández, Gustavo A. and Pellegrino, Daniel and Seoane-Sepúlveda, Juan B. (2012) The asymptotic growth of the constants in the Bohnenblust-Hille inequality is optimal. Journal of Functional Analysis, 263 (2). pp. 415-428. ISSN 0022-1236

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Official URL: http://www.sciencedirect.com/science/journal/00221236




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 C-m's growing at an exponential rate of certain power of m.


Item Type:Article
Uncontrolled Keywords:Bohnenblust-Hille inequality; Asymptotic growth; Optimal constants; Absolutely summing operators
Subjects:Sciences > Mathematics > Functional analysis and Operator theory
ID Code:19941
Deposited On:13 Feb 2013 16:25
Last Modified:28 Nov 2016 09:28

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