Lower bounds for the constants in the Bohnenblust–Hille inequality: The case of real scalars
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- by D. Diniz, G. A. Muñoz-Fernández, D. Pellegrino and J. B. Seoane-Sepúlveda PDF
- Proc. Amer. Math. Soc. 142 (2014), 575-580 Request permission
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 \begin{equation*} \left ( \sum \limits _{i_{1},...,i_{m}=1}^{N}\left \vert T(e_{i_{^{1}}},...,e_{i_{m}})\right \vert ^{\frac {2m}{m+1}}\right ) ^{\frac {m+1}{2m}}\leq C_{m}\left \Vert T\right \Vert \end{equation*} 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 $C_{m}$.References
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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_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
- MR Author ID: 680972
- Email: jseoane@mat.ucm.es
- 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
- © Copyright 2013 American Mathematical Society
- 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
- MathSciNet review: 3133998