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Optimal Hardy-Littlewood type inequalities for polynomials and multilinear operators. (English) Zbl 1342.26040
J. E. Littlewood [Q. J. Math., Oxf. Ser. 1, 164–174 (1930; JFM 56.0335.01)] introduced an inequality for bounded bilinear forms on \(c_0 \times c_0\), which was generalized and improved by H. F. Bohnenblust and E. Hille [Ann. Math. (2) 32, 600–622 (1931; Zbl 0001.26901)] for continuous multilinear forms on \(c_0 \times c_0 \times \cdots \times c_0\), by G. H. Hardy and J. E. Littlewood [Q. J. Math., Oxf. Ser. 5, 241–254 (1934; JFM 60.0335.01)] for continuous multilinear forms on \(l_{p_1} \times l_{p_2} \times \cdots \times l_{p_m}\), by A. Defant and P. Sevilla-Peris [J. Funct. Anal. 256, No. 5, 1642–1664 (2009; Zbl 1171.46034)] for \(l_s\)-valued continuous multilinear mappings on \(c_0 \times c_0 \times \cdots \times c_0\) and by N. Albuquerque et al. [J. Funct. Anal. 266, No. 6, 3726–3740 (2014; Zbl 1319.46035)] and V. Dimant and P. Sevilla-Peris [“Summation of coefficients of polynomials on \(l_p\)-spaces”, Preprint, arXiv:1309.6063] for \(l_s\)-valued continuous multilinear mappings on \(l_{p_1} \times l_{p_2} \times \cdots \times l_{p_m}\).
The authors study in depth the remaining cases of the Defant and Sevilla-Peris result [loc. cit.] and give a simpler proof of the sufficient part of the Defant and Sevilla-Peris theorem to prove a more general result. Moreover, the authors prove similar results for multilinear mappings with arbitrary codomains which assume their cotypes and obtain better estimates for the constants of vector-valued Bohnenblust-Hille inequalities [loc. cit.].
The techniques and the ideas are simple and elementary and the paper is systematically well-organized.

MSC:
26D07 Inequalities involving other types of functions
26D15 Inequalities for sums, series and integrals
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