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Absolutely summing multilinear operators via interpolation. (English) Zbl 1331.46036
In [J. Funct. Anal. 266, No. 6, 3726-3740 (2014; Zbl 1319.46035)], N. Albuquerque et al. used an interpolative technique to prove the sharpness of a family of inequalities of which the multilinear Bohnenblust-Hille inequality is a particular case. In this paper, the authors introduce a variation of a class of multiple summing operators and, using the technique mentioned above, prove a more general inclusion result that encompasses other known ones and allows to recover the more recent estimates of the multilinear Bohnenblust-Hille constants. Among other possible applications, their main result also gives information about the growth of the constants of variants of the Bohnenblust-Hille inequality introduced in [D. Nuñez-Alarcón et al., J. Funct. Anal. 264, No. 1, 326–336 (2013; Zbl 1264.46032)].

MSC:
46G25 (Spaces of) multilinear mappings, polynomials
47L22 Ideals of polynomials and of multilinear mappings in operator theory
47H60 Multilinear and polynomial operators
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[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) · Zbl 1319.46035
[2] 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) · Zbl 1331.46037
[3] Benedek, A.; Panzone, R., The space \(L_p\), with mixed norm, Duke Math. J., 28, 301-324, (1961) · Zbl 0107.08902
[4] 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
[5] Bohnenblust, H.; Hille, E., On the absolute convergence of Dirichlet series, Ann. of Math. (2), 32, 3, 600-622, (1931) · JFM 57.0266.05
[6] Bombal, F.; Peréz-García, D.; Villanueva, I., Multilinear extensions of Grothendieck’s theorem, Q. J. Math., 55, 441-450, (2004) · Zbl 1078.46030
[7] Botelho, G.; Braunss, H.-A.; Junek, H.; Pellegrino, D., Inclusions and coincidences for multiple summing multilinear mappings, Proc. Amer. Math. Soc., 137, 3, 991-1000, (2009) · Zbl 1175.46037
[8] Botelho, G.; Michels, C.; Pellegrino, D., Complex interpolation and summability properties of multilinear operators, Rev. Mat. Complut., 23, 139-161, (2010) · Zbl 1210.46031
[9] Botelho, G.; Pellegrino, D., When every multilinear mapping is multiple summing, Math. Nachr., 282, 10, 1414-1422, (2009) · Zbl 1191.46041
[10] Defant, A.; Popa, D.; Schwarting, U., Coordinatewise multiple summing operators in Banach spaces, J. Funct. Anal., 259, 1, 220-242, (2010) · Zbl 1205.46026
[11] Diestel, J.; Jarchow, H.; Tonge, A., Absolutely summing operators, Cambridge Stud. Adv. Math., vol. 43, (1995), Cambridge University Press Cambridge · Zbl 0855.47016
[12] Diniz, D.; Muñoz-Fernádez, 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., 142, 2, 575-580, (2014) · Zbl 1291.46040
[13] Garling, D. J.H., Inequalities: A journey into linear analysis, (2007), Cambridge University Press Cambridge · Zbl 1135.26014
[14] Haagerup, U., The best constants in the Khintchine inequality, Studia Math., 70, 3, 231-283, (1981), (1982) · Zbl 0501.46015
[15] König, H., On the best constants in the Khintchine inequality for Steinhaus variables, Israel J. Math., 203, 1, 23-57, (2014) · Zbl 1314.46017
[16] Matos, M. C., Fully absolutely summing mappings and Hilbert Schmidt operators, Collect. Math., 54, 111-136, (2003) · Zbl 1078.46031
[17] Mitiagin, B. S.; Pełczyński, A., Nuclear operators and approximative dimensions, (Proceedings of the International Congress of Mathematicians, Moscow, (1966))
[18] Montanaro, A., Some applications of hypercontractive inequalities in quantum information theory, J. Math. Phys., 53, (2012) · Zbl 1278.81045
[19] Núñez-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
[20] Núñez-Alarcón, D.; Pellegrino, D.; Seoane-Sepúlveda, J. B.; Serrano-Rodríguez, D. M., There exist multilinear Bohnenblust-Hille constants \((C_n)_{n \in \mathbb{N}}\) with \(\lim_{n \rightarrow \infty}(C_{n + 1} - C_n) = 0\), J. Funct. Anal., 264, (2013) · Zbl 1264.46033
[21] Pérez-García, D.; Villanueva, I. V., Multiple summing operators on Banach spaces, J. Math. Anal. Appl., 285, 1, 86-96, (2003) · Zbl 1044.46037
[22] Pietsch, A., Absolut p-summierende abbildungen in normierten raümen, Studia Math., 28, 333-353, (1967) · Zbl 0156.37903
[23] Pietsch, A., Ideals of multilinear functionals, (Proceedings of the Second International Conference on Operator Algebras, Ideals and Their Applications in Theoretical Physics, Leipzig, Teubner-Texte Math., (1983)), 185-199
[24] Popa, D.; Sinnamon, G., Blei’s inequality and coordinatewise multiple summing operators, Publ. Mat., 57, 455-475, (2013) · Zbl 1286.26017
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