[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Log in

On the generalized Bohnenblust–Hille inequality for real scalars

  • Published:
Positivity Aims and scope Submit manuscript

Abstract

In this paper we obtain new lower and upper estimates for the sharp constants in the generalized Bohnenblust–Hille inequality introduced in Albuquerque et al. (J Funct Anal 266:3726–3740, 2014). We apply these results to find optimal constants in the generalized Bohnenblust–Hille inequality and also to recover the optimal constants of the mixed \(\left( \ell _{1},\ell _{2}\right) \)-Littlewood inequalities recently obtained in Pellegrino (J Number Theory 160:11–18, 2016) and Pellegrino and Teixeira (Commun Contemp Math, to appear).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Fig. 1

Similar content being viewed by others

References

  1. Albuquerque, N., Bayart, F., Pellegrino, D., Seoane-Sepúlveda, J.B.: Sharp generalizations of the multilinear Bohnenblust–Hille inequality. J. Funct. Anal. 266, 3726–3740 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Araújo, G., Pellegrino, D.: On the constants of the Bohnenblust-Hille inequality and Hardy-Littlewood inequalities. Bull. Braz. Math. Soc. (2016). doi:10.1007/s00574-016-0016-6

    MATH  Google Scholar 

  3. Bayart, F., Pellegrino, D., Seoane-Sepúlveda, J.B.: The Bohr radius of the \(n\)-dimensional polydisk is equivalent to \(\sqrt{(\log n)/n}\). Adv. Math. 264, 726–746 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bohnenblust, H.F., Hille, E.: On the absolute convergence of Dirichlet series. Ann. Math. 32, 600–622 (1931)

    Article  MathSciNet  MATH  Google Scholar 

  5. Davie, A.M.: Quotient algebras of uniform algebras. J. Lond. Math. Soc. 7, 31–40 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  6. Diniz, D., Muñoz-Fernández, G.A., Pellegrino, D., Seoane-Sepúlveda, J.B.: Lower bounds for the constants in the Bohnenblust–Hille inequality: the case of real scalars. Proc. Am. Math. Soc. 142(2), 575–580 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Garling, D.J.H.: Inequalities: A Journey into Linear Analysis. Cambridge University Press, Cambridge (2007)

    Book  MATH  Google Scholar 

  8. Haagerup, U.: The best constants in the Khinchine inequality. Stud. Math. 70, 231–283 (1982)

    Article  MATH  Google Scholar 

  9. Kaijser, S.: Some results in the metric theory of tensor products. Stud. Math. 63(2), 157–170 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  10. Montanaro, A.: Some applications of hypercontractive inequalities in quantum information theory. J. Math. Phys. 53, 122206 (2012). doi:10.1063/1.4769269

    Article  MathSciNet  MATH  Google Scholar 

  11. Núñez-Alarcón, D., Pellegrino, D., Seoane-Sep úlveda, J.B., Serrano-Rodríguez, D.M.: There exist multilinear Bohnenblust–Hille constants \(\left( C_{n}\right) _{n=1}^{\infty }\) with \( \lim _{n\rightarrow \infty }\left( C_{n+1}-C_{n}\right) =0\), J. Funct. Anal. 264, 429–463 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Pellegrino, D.: The optimal constants of the mixed \(\left( \ell _{1},\ell _{2}\right) \)-Littlewood inequality. J. Number Theory 160, 11–18 (2016)

    Article  MathSciNet  Google Scholar 

  13. Pellegrino, D., Teixeira, E.: Towards sharp Bohnenblust–Hille constants. Commun. Contemp. Math. (to appear)

  14. Queffélllec, H.: Bohr’s vision of ordinary Dirichlet series: old and new results. J. Anal. 3, 43–60 (1995)

    MathSciNet  Google Scholar 

  15. Santos, J., Velanga, T.: On the Bohnenblust-Hille inequality for multilinear forms. Results Math. (2016). doi:10.1007/s00025-016-0628-6

    MATH  Google Scholar 

Download references

Acknowledgements

The authors are indebted to the anonymous referee for his/her important contributions to the final version of this paper.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Daniel Núñez-Alarcón.

Additional information

This paper is dedicated to Julieta.

N. Caro is supported by FACEPE Grant APQ-0892-1.01/14, D. Núñez and D. M. Serrano are supported by Capes Grants 000785/2015-06 and 000786/2015-02, respectively.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Caro, N., Núñez-Alarcón, D. & Serrano-Rodríguez, D.M. On the generalized Bohnenblust–Hille inequality for real scalars. Positivity 21, 1439–1455 (2017). https://doi.org/10.1007/s11117-017-0478-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11117-017-0478-9

Keywords

Mathematics Subject Classification

Navigation