On the Bohnenblust-Hille inequality and a variant of Littlewood's 4/3 inequality

dc.contributor.authorNuñez Alarcón, D
dc.contributor.authorPellegrino, Daniel
dc.contributor.authorSeoane-Sepúlveda, Juan B.
dc.description.abstractThe search for sharp constants for inequalities of the type Littlewood's 4/3 and Bohnenblust-Hille has lately shown unexpected applications in many fields such as Analytic Number Theory, Quantum Information Theory, or in results on n-dimensional Bohr radii. Recent estimates obtained for the multilinear Bohnenblust-Hille inequality (for real scalars) have been used, as a crucial tool, by A. Montanaro in order to solve problems in Quantum XOR games. Here, among other results, we obtain new upper bounds for the Bohnenblust-Hille constants (for complex scalars). For bilinear forms, we provide optimal constants of variants of Littlewood's 4/3 inequality (for real scalars) when the exponent 4/3 is replaced by any r >= 4/3. We also prove that the optimal constants in real case are always strictly greater than those from the complex case.
dc.description.departmentDepto. de Análisis Matemático y Matemática Aplicada
dc.description.facultyFac. de Ciencias Matemáticas
dc.description.facultyInstituto de Matemática Interdisciplinar (IMI)
dc.description.sponsorshipMinistry of Science and Innovation
dc.identifier.citationS. Aaronson, A. Ambainis, The Need for Structure in Quantum Speedups, Electron. Colloq. Comput. Complex.,vol. 110, 2009. A. Baernstein, R.C. Culverhouse, Majorization of sequences, sharp vector Khinchin inequalities, and bisubharmonic functions, Studia Math. 152 (3) (2002) 231–248. H.P. Boas, D. Khavinson, Bohr’s power series theorem in several variables, Proc. Amer. Math. Soc. 125 (1997)2975–2979. H.F. Bohnenblust, E. Hille, On the absolute convergence of Dirichlet series, Ann. of Math. (2) 32 (3) (1931) 600–622. A. Defant, L. Frerick, J. Ortega-Cerdà, M. Ounaïes, K.Seip, The Bohnenblust–Hille inequality for homogeneous polynomials is hypercontractive, Ann. of Math. (2) 174 (1)(2011) 485–497. A. Defant, D. Popa, U. Schwarting, Coordinatewise multiple summing operators in Banach spaces, J. Funct.Anal. 259 (1) (2010) 220–242. A. Defant, P. Sevilla-Peris, A new multilinear insight on Littlewood’s 4/3-inequality, J. Funct. Anal. 256 (5) (2009)1642–1664. J. Diestel, H. Jarchow, A. Tonge, Absolutely Summing Operators, Cambridge Stud. Adv. Math., vol. 43, Cambridge University Press, Cambridge, 1995. D. Diniz, G.A. Muñoz-Fernández, D. Pellegrino, J.B. Seoane-Sepúlveda, The asymptotic growth of the constants in the Bohnenblust–Hille inequality is optimal, J. Funct. Anal. 263 (2) (2012) 415–428. D. Diniz, G.A. Muñoz-Fernández, D. Pellegrino, J.B. Seoane-Sepúlveda, Lower bounds for the constants in the Bohnenblust–Hille inequality: the case of real scalars,Proc. Amer. Math. Soc., in press. D.J.H. Garling, Inequalities: A Journey Into Linear Analysis, Cambridge University Press, Cambridge, 2007. U. Haagerup, The best constants in the Khintchine inequality, Studia Math. 70 (3) (1981) 231–283 (1982). M. Junge, C. Palazuelos, D. Pérez-García, I. Villanueva,M.M. Wolf, Unbounded violations of bipartite Bell inequalities via operator space theory, Comm. Math. Phys. 300 (3) (2010) 715–739. S. Kaijser, Some results in the metric theory of tensor products, Studia Math. 63 (2) (1978) 157–170. J.E. Littlewood, On bounded bilinear forms in an infinite number of variables, Q. J. Math. 1 (1930) 164–174. A. Montanaro, Some applications of hypercontractive inequalities in quantum information theory,arXiv:1208.0161v2 [quant-ph]. G.A. Muñoz-Fernández, D. Pellegrino, J.B. Seoane-Sepúlveda, Estimates for the asymptotic behavior of the constants in the Bohnenblust–Hille inequality, Linear Multilinear Algebra 60 (2012) 573–582. D. Nuñez-Alarcón, D. Pellegrino, On the growth of the optimal constants of the multilinear Bohnenblust–Hille inequality, arXiv:1205.2385v1 [math.FA]. D. Nuñez-Alarcón, D. Pellegrino, J.B. Seoane-Sepúlveda,D.M. Serrano-Rodríguez, There exist multilinear Bohnenblust–Hille constants (Cn)∞ n=1 with limn Cn+1 −Cn = 0,arXiv:1207.0124 [math.FA]. A. Pełczy´nski, Norms of classical operators in function spaces, in: Colloquium in Honor of Laurent Schwartz, vol. 1,Palaiseau, 1983, Astérisque 131 (1985) 137–162. D. Pellegrino, J.B. Seoane-Sepúlveda, New upper bounds for the constants in the Bohnenblust–Hille inequality,J. Math. Anal. Appl. 386 (1) (2012) 300–307. H. Queffélec, H. Bohr’s vision of ordinary Dirichlet series: old and new results, J. Anal. 3 (1995) 43–60. J. Sawa, The best constant in the Khinchine inequality for complex Steinhaus variables, the case p = 1, Studia Math. 81 (1) (1985) 107–126.
dc.journal.titleJournal of Functional Analysis
dc.rights.accessRightsrestricted access
dc.subject.keywordBohnenblust–Hille Theorem
dc.subject.keywordLittlewood’s 4/3 inequality
dc.subject.keywordSteinhaus random variables
dc.subject.ucmFísica matemática
dc.titleOn the Bohnenblust-Hille inequality and a variant of Littlewood's 4/3 inequality
dc.typejournal article
Original bundle
Now showing 1 - 2 of 2
No Thumbnail Available
143.36 KB
Adobe Portable Document Format
Thumbnail Image
165.25 KB
Adobe Portable Document Format