Abstract
In this paper we obtain quite general and definitive forms for Hardy-Littlewood type inequalities. Moreover, when restricted to the original particular cases, our approach provides much simpler and straightforward proofs and we are able to show that in most cases the exponents involved are optimal. The technique we used is a combination of probabilistic tools and of an interpolative approach; this former technique is also employed in this paper to improve the constants for vector-valued Bohnenblust-Hilletype inequalities.
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N. Albuquerque, F. Bayart, D. Pellegrino and J. B. Seoane-Sepúlveda, Sharp generalizations of the multilinear Bohnenblust-Hille inequality, Journal of Functional Analysis 266 (2014), 3726–3740.
F. Bayart, D. Pellegrino and J. B. Seoane-Sepúlveda, The Bohr radius of the ndimensional polydisk is equivalent to \(\sqrt {(\log n)/n} \), Advances in Mathematics 264 (2014), 726–746.
G. Bennett, Schur multipliers, Duke Mathematical Journal 44 (1977), 603–639.
J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Grundlehren der Mathematischen Wissenschaften, Vol. 223, Springer-Verlag, Berlin, 1976.
H. P. Boas, Majorant series, Journal of the Korean Mathematical Society 37 (2000), 321–337.
H. Bohnenblust and E. Hille, On the absolute convergence of Dirichlet series, Annals of Mathematics 32, (1931), 600–622.
F. Bombal, D. Pérez-García and I. Villanueva, Multilinear extensions of Grothendieck’s theorem, The Quarterly Journal 0f Mathematics 55 (2004), 441–450.
G. Botelho, Cotype and absolutely summing multilinear mappings and homogeneous polynomials, Proceedings of the Royal Irish Academy. Section A. Mathematical and Physical Sciences 97 (1997), 145–153.
B. Carl, Absolut-(p, 1)-summierende identische Operatoren von l u in l v, Mathematische Nachrichten 63 (1974), 353–360.
A. Defant, D. Popa and U. Schwarting, Coordinatewise multiple summing operators in Banach spaces, Journal of Functional Analysis 259 (2010), 220–242.
A. Defant and P. Sevilla-Peris, A new multilinear insight on Littlewood’s 4/3-inequality, Journal of Functional Analysis 256 (2009), 1642–1664.
A. Defant and P. Sevilla-Peris, The Bohnenblust-Hille cycle of ideas from a modern point of view, Functiones et Approximatio Commentarii Mathematici 50 (2014), 55–127.
J. Diestel, H. Jarchow and A. Tonge, Absolutely Summing Operators, Cambridge Studies in Advanced Mathematics, Vol. 43, Cambridge University Press, Cambridge, 1995.
V. Dimant and P. Sevilla-Peris, Summation of coefficients of polynomials on ℓp spaces, arXiv:1309.6063v1 [math.FA] 24 Sep 2013.
J. F. Fournier, Mixed norms and rearrangements: Sobolev’s inequality and Littlewood’s inequality, Annali di Matematica Pura ed Applicata 148 (1987), 51–76.
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 Mathematica 70 (1981), 231–283.
G. Hardy and J. Littlewood, Bilinear forms bounded in space [p, q], Quarterly Journal of Mathematics 5 (1934), 241–254.
G. Hardy, J. Littlewood and G. Polya, Inequalities, Second edition, Cambridge at the University press, 1952.
H. König and S. Kwapień, Best Khintchine type inequalities for sums of independent, rotationally invariant random vectors, Positivity 5 (2001), 115–152.
S. Kwapień, Some remarks on (p, q)-absolutely summing operators in ℓp-spaces, Studia Mathematica 29 (1968), 327–337.
J. E. Littlewood, On bounded bilinear forms in an infinite number of variables, Quarterly Journal of Mathematcis 1 (1930), 164–174.
M. C. Matos, Fully absolutely summing mappings and Hilbert-Schmidt multilinear mappings, Collectanea Mathematica 54 (2003), 111–136.
B. Maurey and G. Pisier, Séries de variables aleatoires vectorielles indépendantes et proprietés géometriques des espaces de Banach, Studia Mathematica 58 (1976), 45–90.
D. Pérez-García and I. Villanueva, Multiple summing operators on Banach spaces, Journal of Mathematical Analysis and Applications 285 (2003), 86–96.
D. Pérez-García and I. Villanueva, Multiple summing operators on C(K)spaces, Arkiv för Mathematik 42 (2004), 153–171.
T. Praciano-Pereira, On bounded multilinear forms on a class of ℓp spaces, Journal of Mathematical Analysis and Applications 81 (1981), 561–568.
U. C. Schwarting, Vector Valued Bohnenblust-Hille Inequalities, Thesis, 2013.
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N. Albuquerque was supported by PDSE/CAPES 12038-13-0.
D. Pellegrino and J. B. Seoane-Sepúlveda were supported by CNPq Grant 401735/2013-3 (PVE - Linha 2).
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Albuquerque, N., Bayart, F., Pellegrino, D. et al. Optimal Hardy-Littlewood type inequalities for polynomials and multilinear operators. Isr. J. Math. 211, 197–220 (2016). https://doi.org/10.1007/s11856-015-1264-7
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DOI: https://doi.org/10.1007/s11856-015-1264-7