A new multilinear insight on the classical Littlewood's 4/3-inequality

[EN] We unify Littlewood's classical 4/3-inequality (a forerunner of Grothendieck's inequality) together with its in-linear extension due to Bohnenblust and Hille (which originally settled Bohr's absolute convergence problem for Dirichlet series) with a scale of inequalties of Bennett and Carl in l(p)-spaces (which are of fundamental importance in the theory of eigenvalue distribution of power compact operators). As an application we give estimates for the monomial coefficients of homogeneous l(p)-valued polynomials on c(0). (C) 2008 Elsevier Inc. All rights reserved.Supported by the MEC Project MTM2005-08210. Supported by the MEC Project MTM2005-08210 and grants PR2007-0384 (MEC) and UPV-PAID-00-07.Defant, A.; Sevilla Peris, P. (2009). A new multilinear insight on the classical Littlewood's 4/3-inequality. Journal of Functional Analysis. 5(256):1642-1664. https://doi.org/10.1016/j.jfa.2008.07.005164216645256Bennett, G. (1973). Inclusion mappings between lp spaces. Journal of Functional Analysis, 13(1), 20-27. doi:10.1016/0022-1236(73)90064-5Bohnenblust, H. F., & Hille, E. (1931). On the Absolute Convergence of Dirichlet Series. The Annals of Mathematics, 32(3), 600. doi:10.2307/1968255Bombal, F. (2004). Multilinear extensions of Grothendieck’s theorem. The Quarterly Journal of Mathematics, 55(4), 441-450. doi:10.1093/qmath/hah017Bonet, J., & Peris, A. (1991). On the injective tensor product of quasinormable spaces. Results in Mathematics, 20(1-2), 431-443. doi:10.1007/bf03323184G. Botelho, H.-A. Braunss, H. Junek, D. Pellegrino, Inclusions and coincidences for multiple multilinear mappings, preprintCarl, B. (1974). Absolut-(p , 1)-summierende identische Operatoren von l u in l v. Mathematische Nachrichten, 63(1), 353-360. doi:10.1002/mana.3210630132Choi, Y. s. (2001). Estimates for absolutely summing norms of polynomials and multilinear maps. The Quarterly Journal of Mathematics, 52(1), 1-12. doi:10.1093/qjmath/52.1.1Davie, A. M. (1973). Quotient Algebras of Uniform Algebras. Journal of the London Mathematical Society, s2-7(1), 31-40. doi:10.1112/jlms/s2-7.1.31Defant, A., & Maestre, M. (1994). Property (BB) and holomorphic functions on Fréchet-Montel spaces. Mathematical Proceedings of the Cambridge Philosophical Society, 115(2), 305-313. doi:10.1017/s0305004100072108A. Defant, C. Prengel, Volume estimates in spaces of homogeneous polynomials, Math. Z., doi:10.1007/s00209-008-0358-xA. Defant, P. Sevilla-Peris, Convergence of Dirichlet polynomials in Banach spaces, preprintDefant, A., Dı́az, J. C., Garcı́a, D., & Maestre, M. (2001). Unconditional Basis and Gordon–Lewis Constants for Spaces of Polynomials. Journal of Functional Analysis, 181(1), 119-145. doi:10.1006/jfan.2000.3702Defant, A., Maestre, M., & Sevilla-Peris, P. (2002). Cotype 2 estimates for spaces of polynomials on sequence spaces. Israel Journal of Mathematics, 129(1), 291-315. doi:10.1007/bf02773168A. Defant, D. García, M. Maestre, D. Pérez-García, Bohr's strip for vector valued Dirichlet series, Math. Ann., doi:10.1007/s00208-008-0246-zDineen, S. (1999). Complex Analysis on Infinite Dimensional Spaces. Springer Monographs in Mathematics. doi:10.1007/978-1-4471-0869-6Kaisjer, S. (1978). Some results in the metric theory of tensor products. Studia Mathematica, 63(2), 157-170. doi:10.4064/sm-63-2-157-170Kwapień, S. (1968). Some remarks on (p,q)-absolutely summing operators in $l_{p}$-spaces. Studia Mathematica, 29(3), 327-337. doi:10.4064/sm-29-3-327-337LITTLEWOOD, J. E. (1930). ON BOUNDED BILINEAR FORMS IN AN INFINITE NUMBER OF VARIABLES. The Quarterly Journal of Mathematics, os-1(1), 164-174. doi:10.1093/qmath/os-1.1.164Maurey, B., & Pisier, G. (1976). Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach. Studia Mathematica, 58(1), 45-90. doi:10.4064/sm-58-1-45-90Orlicz, W. (1933). Über unbedingte Konvergenz in Funktionenraümen (I). Studia Mathematica, 4(1), 33-37. doi:10.4064/sm-4-1-33-37Pérez-Garcı́a, D., & Villanueva, I. (2003). Multiple summing operators on Banach spaces. Journal of Mathematical Analysis and Applications, 285(1), 86-96. doi:10.1016/s0022-247x(03)00352-4Praciano-Pereira, T. (1981). On bounded multilinear forms on a class of lp spaces. Journal of Mathematical Analysis and Applications, 81(2), 561-568. doi:10.1016/0022-247x(81)90082-2Sawa, J. (1985). The best constant in the Khintchine inequality for complex Steinhaus variables, the case p=1. Studia Mathematica, 81(1), 107-126. doi:10.4064/sm-81-1-107-126Talagrand, M. (1992). Cotype and (q, 1)-summing norm in a Banach space. Inventiones Mathematicae, 110(1), 545-556. doi:10.1007/bf01231344Tonge, A. (1984). Polarization and the two-dimensional Grothendieck inequality. Mathematical Proceedings of the Cambridge Philosophical Society, 95(2), 313-318. doi:10.1017/s030500410006157