zbMATH — the first resource for mathematics

Sharp generalizations of the multilinear Bohnenblust-Hille inequality. (English) Zbl 1319.46035
Let \(\mathbb{K}=\mathbb{R}\) or \(\mathbb{C}\). If \(A\) is a continuous \(m\)-linear form on \(c_{0}\times\cdots\times c_{0}\), for \(m \geq 2\), then there is a constant \(C_{m}\) (depending only on \(m\)) such that \[ \left( \sum_{i_{1},\dots,i_{m}=1}^{\infty} \left| A(e_{i_{1}},\dots,e_{i_{m}})\right| ^{\frac{2m}{m+1}}\right) ^{\frac{m+1}{2m}}\leq C_{m}\left\| A\right\| . \] This is the Bohnenblust-Hille inequality and, when \(m=2\), we have the well-known Littlewood \(4/3\) inequality. During a few decades, some generalizations of the Bohnenblust-Hille inequality have been presented including, as quoted by authors, those that appear in the papers of A. Defant and P. Sevilla-Peris [J. Funct. Anal. 256, No. 5, 1642–1664 (2009; Zbl 1171.46034)] and T. Praciano-Pereira [J. Math. Anal. Appl. 81, 561–568 (1981; Zbl 0497.46007)].
In the main theorem of this interesting paper, the authors show that the Bohnenblust-Hille inequality is a particular case of a more general and sharp family of inequalities and that the above mentioned generalizations can be adapted to their approach, that is, can be recovered with appropriate choices of parameters and objects involved. The paper also attracts attention by the use of economical and/or simple arguments in many proofs.

46G25 (Spaces of) multilinear mappings, polynomials
47L22 Ideals of polynomials and of multilinear mappings in operator theory
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
Full Text: DOI arXiv
[1] Bennett, G., Inclusion mappings between \(l^p\) spaces, J. Funct. Anal., 13, 20-27, (1973) · Zbl 0255.47033
[2] Bergh, J.; Löfström, J., Interpolation spaces. an introduction, Grundlehren Math. Wiss., vol. 223, (1976), Springer-Verlag Berlin · Zbl 0344.46071
[3] Blasco, O.; Botelho, G.; Pellegrino, D.; Rueda, P., Summability of multilinear mappings: Littlewood, Orlicz and beyond, Monatsh. Math., 163, 2, 131-147, (2011) · Zbl 1246.46045
[4] Boas, H. P., The football player and the infinite series, Notices Amer. Math. Soc., 44, 11, 1430-1435, (1997) · Zbl 0909.30001
[5] Boas, H. P., Majorant series, J. Korean Math. Soc., 37, 321-337, (2000) · Zbl 0965.32001
[6] Bohnenblust, H. F.; Hille, E., On the absolute convergence of Dirichlet series, Ann. of Math. (2), 32, 3, 600-622, (1931) · JFM 57.0266.05
[7] Carl, B., Absolut-\((p, 1)\)-summierende identische operatoren von \(l_u\) in \(l_v\), Math. Nachr., 63, 353-360, (1974), (in German) · Zbl 0292.47019
[8] Defant, A.; Sevilla-Peris, P., A new multilinear insight on Littlewood’s 4/3-inequality, J. Funct. Anal., 256, 5, 1642-1664, (2009) · Zbl 1171.46034
[9] Diniz, D.; Muñoz-Fernández, G. A.; Pellegrino, D.; Seoane-Sepúlveda, J. B., The asymptotic growth of the constants in the Bohnenblust-Hille inequality is optimal, J. Funct. Anal., 263, 2, 415-428, (2012) · Zbl 1252.46034
[10] 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. Amer. Math. Soc., (2013), in press · Zbl 1291.46040
[11] Littlewood, J. E., On bounded bilinear forms in an infinite number of variables, Quart. J. Math. Oxford, 1, 164-174, (1930) · JFM 56.0335.01
[12] Montanaro, A., Some applications of hypercontractive inequalities in quantum information theory, J. Math. Phys., 53, (2012) · Zbl 1278.81045
[13] Nuñez-Alarcón, D.; Pellegrino, D.; Seoane-Sepúlveda, J. B., On the Bohnenblust-Hille inequality and a variant of Littlewood’s 4/3 inequality, J. Funct. Anal., 264, 326-336, (2013) · Zbl 1264.46032
[14] Nuñez-Alarcón, D.; Pellegrino, D.; Seoane-Sepúlveda, J. B.; Serrano-Rodríguez, D. M., There exist multilinear Bohnenblust-Hille constants \((C_n)_{n = 1}^\infty\) with \(\lim_{n \rightarrow \infty}(C_{n + 1} - C_n) = 0\), J. Funct. Anal., 264, 2, 429-463, (2013) · Zbl 1264.46033
[15] Praciano-Pereira, T., On bounded multilinear forms on a class of \(l^p\) spaces, J. Math. Anal. Appl., 81, 2, 561-568, (1981) · Zbl 0497.46007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.