Muñoz-Fernández, G.A. and Pellegrino, D. and Seoane Sepúlveda, Juan Benigno
(2012)
*Estimates for the asymptotic behaviour of the constants in the Bohnenblust-Hille inequality.*
Linear & Multilinear Algebra, 60
(5).
pp. 573-582.
ISSN 0308-1087

PDF
Restricted to Repository staff only until 2020. 144kB |

Official URL: http://www.tandfonline.com/doi/pdf/10.1080/03081087.2011.613833

## Abstract

A classical inequality due to H. F. Bohnenblust and E. Hille states that for every positive integer n there is a constant C-n > 0 so that (Sigma(N)(i1,...,in=1) vertical bar U(e(i1), . . . , e(in))vertical bar(2n/n+1))(n+1/2n) <= C-n parallel to U parallel to for every positive integer N and every n-linear mapping U : l(infinity)(N) x . . x l(infinity)(N) -> C. The original estimates for those constants from Bohnenblust and Hille are C-n = n(n+1/2n)2(n-1/2). In this note we present explicit formulae for quite better constants, and calculate the asymptotic behaviour of these estimates, completing recent results of the second and third authors. For example, we show that, if C-R,C- (n) and C-C,C- (n) denote (respectively) these estimates for the real and complex Bohnenblust-Hille inequality then, for every even positive integer n, C-R,C-n/root pi = CC, n/root 2 = 2(n+2/8) . r(n) for a certain sequence {r(n)} which we estimate numerically to belong to the interval (1, 3/2) (the case n odd is similar). Simultaneously, assuming that {r(n)} is in fact convergent, we also conclude that lim(n ->infinity) C-R,C- n/C-R,C- n-1 = lim(n ->infinity) C-C,C- n/C-C,C- n-1 = 2(1/8).

Item Type: | Article |
---|---|

Uncontrolled Keywords: | Bohnenblust–Hille Inequality; Asymptotic growth Classification : |

Subjects: | Sciences > Mathematics > Functional analysis and Operator theory |

ID Code: | 16876 |

References: | N. Alon, K. Makarychev, Y. Makarychev, and A. Naor, Quadratic forms on graphs, Invent. Math. 163 (2006), pp. 499–522. H.F. Bohnenblust and E. Hille, On the absolute convergence of Dirichlet series, Ann Math. 32 (1931), pp. 600–622. G. Botelho, H.-A. Braunss, H. Junek, and D. Pellegrino, Inclusions and coincidences for multiple summing multilinear mappings, Proc. Amer. Math. Soc. 137 (2009), pp. 991–1000. M. Braverman, K. Makarychev, Y. Makarychev, and A. Naor, The Grothendieck constant is strictly smaller than Krivine’s bound (2011). Available at arXiv:1103.6161v2. A.M. Davie, Quotient algebras of uniform algebras, J. London Math. Soc. 7 (1973), pp. 31–40. A. Defant, L. Frerick, J. Ortega-Cerdá, M. Ounaïes, and K. Seip, The Bohnenblust–Hille inequality for homogeneous polynomials is hypercontractive, Ann. Math. 174 (2011), pp. 485–497. A. Defant, D. Popa, and U. Schwarting, Coordinatewise multiple summing operators in Banach spaces (English summary), J. Funct. Anal. 259 (2010), pp. 220–242. A. Defant and P. Sevilla-Peris, A new multilinear insight on Littlewood’s 4/3-inequality, J. Funct. Anal. 256 (2009), pp. 1642–1664. J. Diestel, J. Fourie, and J. Swart, The Metric Theory of Tensor Products, American Mathematical Society, Providence, RI, 2008. J. Diestel, H. Jarchow, and A. Tonge, Absolutely Summing Operators (English Summary), Cambridge Studies in Advanced Mathematics, Vol. 43, Cambridge University Press,Cambridge, 1995. P.C. Fishburn and J.A. eeds,Bellinequalities,Grothendieck’s constant, and root two, IAM J. Discrete Math. 7 (1994), pp. 48–56. A. Grothendieck, Résume´ de la théorie métrique des produits tensoriels topologiques French) [Summary of the metric theory of topological tensor products], Bol. Soc. Mat. ao Paulo 8 (1953), pp. 1–79 (Reprint). U. Haagerup, The best constants in the Khintchine inequality, Stud. Math. 70 (1981), p. 231–283, 1982. S. Kaijser, Some results in the metric theory of tensor products, Stud. Math. 63 (1978), p. 157–170. J. Lindenstrauss and A. Pelczynski , Absolutely summing operators in Lp-spaces and their applications, Stud. Math. 29 (1968), pp. 275–326. J.E. Littlewood, On bounded bilinear forms in an infinite number of variables, Q. J. Math 1 (1930), pp. 164–174. D. Pellegrino and J.B. Seoane-Sepúlveda, Improving the constants for real and complex Bohnenblust–Hille inequality, preprint (2010). Available at arXiv 1010.0461v2. H. Queffélec, H. Bohr’s vision of ordinary Dirichlet series; Old and new results, J. Anal. 3(1995), pp. 43–60. |

Deposited On: | 25 Oct 2012 08:08 |

Last Modified: | 07 Feb 2014 09:37 |

Repository Staff Only: item control page