×

zbMATH — the first resource for mathematics

On the upper bounds for the constants of the Hardy-Littlewood inequality. (English) Zbl 1298.26066
Summary: The best known upper estimates for the constants of the Hardy-Littlewood inequality for \(m\)-linear forms on \(\ell_p\) spaces are of the form \((\sqrt{2})^{m - 1}\). We present better estimates which depend on \(p\) and \(m\). An interesting consequence is that if \(p \geq m^2\) then the constants have a subpolynomial growth as \(m\) tends to infinity.

MSC:
26D15 Inequalities for sums, series and integrals
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Albuquerque, N.; Bayart, F.; Pellegrino, D.; Seoane-Sepúlveda, J., Sharp generalizations of the multilinear Bohnenblust-Hille inequality, J. Funct. Anal., 266, 3726-3740, (2014) · Zbl 1319.46035
[2] Albuquerque, N.; Bayart, F.; Pellegrino, D.; Seoane-Sepúlveda, J., Optimal Hardy-Littlewood type inequalities for polynomials and multilinear operators, Israel J. Math., (2014), in press · Zbl 1319.46035
[3] Bayart, F.; Pellegrino, D.; Seoane-Sepúlveda, J. B., The Bohr radius of the n-dimensional polydisc is equivalent to \(\sqrt{(\log n) / n}\), (15 Oct. 2013)
[4] Bohnenblust, H. F.; Hille, E., On the absolute convergence of Dirichlet series, Ann. of Math., 32, 600-622, (1931) · JFM 57.0266.05
[5] Diestel, J.; Jarchow, H.; Tonge, A., Absolutely summing operators, (1995), Cambridge University Press Cambridge · Zbl 0855.47016
[6] Dimant, V.; Sevilla-Peris, P., Summation of coefficients of polynomials on \(\ell_p\) spaces · Zbl 1378.46032
[7] Haagerup, U., The best constants in the khinchine inequality, Studia Math., 70, 231-283, (1982) · Zbl 0501.46015
[8] Hardy, G.; Littlewood, J. E., Bilinear forms bounded in space \([p, q]\), Quart. J. Math., 5, 241-254, (1934) · JFM 60.0335.01
[9] König, H., On the best constants in the Khintchine inequality for variables on spheres, (1998), Universität Kiel, Math. Seminar
[10] 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
[11] Nuñez-Alarcón, D.; Pellegrino, D.; Seoane-Sepúlveda, J. B.; Serrano-Rodriguez, 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, 429-463, (2013) · Zbl 1264.46033
[12] Praciano-Pereira, T., On bounded multilinear forms on a class of \(\ell_p\) spaces, J. Math. Anal. Appl., 81, 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.