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Lower bounds for the constants in the Bohnenblust-Hille inequality: The case of real scalars


Authors: D. Diniz, G. A. Muñoz-Fernández, D. Pellegrino and J. B. Seoane-Sepúlveda
Journal: Proc. Amer. Math. Soc. 142 (2014), 575-580
MSC (2010): Primary 46G25, 47H60
DOI: https://doi.org/10.1090/S0002-9939-2013-11791-0
Published electronically: October 25, 2013
MathSciNet review: 3133998
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Abstract: The Bohnenblust-Hille inequality was obtained in 1931 and (in the case of real scalars) asserts that for every positive integer $ m$ there is a constant $ C_{m}$ so that

$\displaystyle \left ( \sum \limits _{i_{1},...,i_{m}=1}^{N}\left \vert T(e_{i_{... ...{\frac {2m}{m+1}}\right ) ^{\frac {m+1}{2m}}\leq C_{m}\left \Vert T\right \Vert$    

for all positive integers $ N$ and every $ m$-linear mapping $ T:\ell _{\infty }^{N}\times \cdots \times \ell _{\infty }^{N}\rightarrow \mathbb{R}$. Since then, several authors have obtained upper estimates for the values of $ C_{m}$. However, the novelty presented in this short note is that we provide lower (and non-trivial) bounds for $ C_{m}$.

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Additional Information

D. Diniz
Affiliation: Unidade Academica de Matemática e Estatística, Universidade Federal de Campina Grande, Caixa Postal 10044, Campina Grande, 58429-970, Brazil
Email: diogo@dme.ufcg.edu.br

G. A. Muñoz-Fernández
Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Plaza de Ciencias 3, Madrid, 28040, Spain
Email: gustavo$\textunderscore$fernandez@mat.ucm.es

D. Pellegrino
Affiliation: Departamento de Matemática, Universidade Federal da Paraíba, 58.051-900 - João Pessoa, Brazil
Email: pellegrino@pq.cnpq.br, dmpellegrino@gmail.com

J. B. Seoane-Sepúlveda
Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Plaza de Ciencias 3, Madrid, 28040, Spain
Email: jseoane@mat.ucm.es

DOI: https://doi.org/10.1090/S0002-9939-2013-11791-0
Keywords: Absolutely summing operators, Bohnenblust--Hille Theorem, Bohnenblust--Hille inequality
Received by editor(s): November 18, 2011
Received by editor(s) in revised form: March 23, 2012
Published electronically: October 25, 2013
Additional Notes: The second and fourth authors were supported by the Spanish Ministry of Science and Innovation (grant MTM2009-07848)
The third author was supported by CNPq Grant 301237/2009-3, CAPES-NF and INCT-Matemática.
Communicated by: Marius Junge
Article copyright: © Copyright 2013 American Mathematical Society

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