Optimal Hardy–Littlewood inequalities uniformly bounded by a universal constant

Albuquerque, Nacib; Araújo, Gustavo; Maia, Mariana; Nogueira, Tony; Pellegrino, Daniel; Santos, Joedson

doi:10.5802/ambp.371

Albuquerque, Nacib ¹ ; Araújo, Gustavo ² ; Maia, Mariana ^1,³ ; Nogueira, Tony ^1,⁴ ; Pellegrino, Daniel ¹ ; Santos, Joedson ¹

¹ Departamento de Matemática Universidade Federal da Paraíba 58.051-900 - João Pessoa, Brazil.
² Departamento de Matemática Universidade Estadual da Paraíba 58.429-500 - Campina Grande, Brazil.
³ & Dep. de Ciência e Tecnologia Univ. Fed. Rural do Semi–Árido 59.700-000 - Caraúbas, Brazil.
⁴ Dep. de Ciênc. Ex. e Tec. da Info. Univ. Fed. Rural do Semi–Árido 59.515-000 - Angicos, Brazil.

Annales mathématiques Blaise Pascal, Tome 25 (2018) no. 1, pp. 1-20.

Résumé

The $m$ -linear version of the Hardy–Littlewood inequality for $m$ -linear forms on $ℓ_{p}$ spaces and $m < p < 2 m,$ recently proved by Dimant and Sevilla-Peris, asserts that

{(\sum_{\begin{matrix} j_{i} = 1 \\ 1 \leq i \leq m \end{matrix}}^{\infty} {|T (e_{j_{1}}, \dots, e_{j_{m}})|}^{\frac{p}{p - m}})}^{\frac{p - m}{p}} \leq 2^{\frac{m - 1}{2}} sup_{\begin{matrix} ∥x_{i}∥ \leq 1 \\ 1 \leq i \leq m \end{matrix}} |T (x_{1}, \dots, x_{m})|

for all continuous $m$ -linear forms $T : ℓ_{p} \times \dots \times ℓ_{p} \to ℝ$ or $ℂ$ . We prove a technical lemma, of independent interest, that pushes further some techniques that go back to the seminal ideas of Hardy and Littlewood. As a consequence, we show that the inequality above is still valid with $2^{(m - 1) / 2}$ replaced by $2^{(m - 1) (p - m) / p}$ . In particular, we conclude that for $m < p \leq m + 1$ the optimal constants of the above inequality are uniformly bounded by $2;$ also, when $m = 2,$ we improve the estimates of the original inequality of Hardy and Littlewood.

Publié le : 2018-07-02

DOI : 10.5802/ambp.371

Classification : 46G25, 47H60
Mots clés : Absolutely summing operators, Hardy–Littlewood inequalities, constants

Affiliations des auteurs :

Albuquerque, Nacib ¹ ; Araújo, Gustavo ² ; Maia, Mariana ^{1, 3} ; Nogueira, Tony ^{1, 4} ; Pellegrino, Daniel ¹ ; Santos, Joedson ¹

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     title = {Optimal {Hardy{\textendash}Littlewood} inequalities uniformly bounded by a universal constant},
     journal = {Annales math\'ematiques Blaise Pascal},
     pages = {1--20},
     publisher = {Universit\'e Clermont Auvergne, Laboratoire de math\'ematiques Blaise Pascal},
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Albuquerque, Nacib; Araújo, Gustavo; Maia, Mariana; Nogueira, Tony; Pellegrino, Daniel; Santos, Joedson. Optimal Hardy–Littlewood inequalities uniformly bounded by a universal constant. Annales mathématiques Blaise Pascal, Tome 25 (2018) no. 1, pp. 1-20. doi : 10.5802/ambp.371. http://archive.numdam.org/articles/10.5802/ambp.371/

Bibliographie
Cité par

[1] Achour, Dahmane; Dahia, Elhadj; Rueda, Pilar; Sánchez-Pérez, Enrique A. Domination spaces and factorization of linear and multilinear summing operators, Quaest. Math., Volume 39 (2016) no. 8, pp. 1071-1092 | DOI | MR

[2] Albuquerque, Nacib; Bayart, Frederic; Pellegrino, Daniel; Seoane-Sepúlveda, Juan B. Sharp generalizations of the multilinear Bohnenblust-Hille inequality, J. Funct. Anal., Volume 266 (2014) no. 6, pp. 3726-3740 | DOI | MR | Zbl

[3] Albuquerque, Nacib; Bayart, Frederic; Pellegrino, Daniel; Seoane-Sepúlveda, Juan B. Optimal Hardy-Littlewood type inequalities for polynomials and multilinear operators, Isr. J. Math., Volume 211 (2016) no. 1, pp. 197-220 | DOI | MR | Zbl

[4] Albuquerque, Nacib; Rezende, Lisiane Anisotropic regularity principle in sequence spaces and applications http://www.worldscientific.com/doi/abs/10.1142/S0219199717500870 (https://www.worldscientific.com/doi/abs/10.1142/S0219199717500870, to appear in Commun. Contemp. Math.) | DOI

[5] Araújo, Gustavo; Pellegrino, Daniel On the constants of the Bohnenblust-Hille and Hardy-Littlewood inequalities, Bull. Braz. Math. Soc. (N.S.), Volume 48 (2017) no. 1, pp. 141-169 | DOI | MR | Zbl

[6] Araújo, Gustavo; Pellegrino, Daniel; da Silva e Silva, Diogo Diniz P. On the upper bounds for the constants of the Hardy-Littlewood inequality, J. Funct. Anal., Volume 267 (2014) no. 6, pp. 1878-1888 | DOI | MR | Zbl

[7] Aron, Richard; Núñez-Alarcón, Daniel; Pellegrino, Daniel; Serrano-Rodríguez, Diana M. Optimal exponents for Hardy-Littlewood inequalities for m-linear operators, Linear Algebra Appl., Volume 531 (2017), pp. 399-422 http://www.sciencedirect.com/science/article/pii/S0024379517303713 | DOI | Zbl

[8] Bayart, Frédéric Multiple summing maps: Coordinatewise summability, inclusion theorems and $p$ -Sidon sets, J. Funct. Anal., Volume 274 (2018) no. 4, pp. 1129-1154 http://www.sciencedirect.com/science/article/pii/S0022123617303270 | DOI | Zbl

[9] Bayart, Frédéric; Pellegrino, Daniel; Seoane-Sepúlveda, Juan B. The Bohr radius of the n-dimensional polydisk is equivalent to $\sqrt{\frac{log n}{n}}$ , Adv. Math., Volume 264 (2014), pp. 726-746 http://www.sciencedirect.com/science/article/pii/S000187081400262X | DOI | Zbl

[10] Bohnenblust, H. Frederic; Hille, Einar On the absolute convergence of Dirichlet series, Ann. Math., Volume 32 (1931) no. 3, pp. 600-622 | DOI | MR | Zbl

[11] Bu, Qingying; Labuschagne, Coenraad C.A. Positive multiple summing and concave multilinear operators on Banach lattices, Mediterr. J. Math., Volume 12 (2015) no. 1, pp. 77-87 | DOI | MR | Zbl

[12] Cavalcante, Wasthenny Some applications of the regularity principle in sequence spaces, Positivity, Volume 22 (2018) no. 1, pp. 191-198 | DOI

[13] Cavalcante, Wasthenny; Núñez-Alarcón, Daniel Remarks on an inequality of Hardy and Littlewood, Quaest. Math., Volume 39 (2016) no. 8, pp. 1101-1113 | DOI | MR

[14] Defant, Andreas; Sevilla-Peris, Pablo A new multilinear insight on Littlewood’s 4/3-inequality, J. Funct. Anal., Volume 256 (2009) no. 5, pp. 1642-1664 | DOI | MR | Zbl

[15] Delgado, Olvido; Sánchez-Pérez, Enrique A. Strong extensions for $q$ -summing operators acting in $p$ -convex Banach function spaces for $1 \leq p \leq q$ , Positivity, Volume 20 (2016) no. 4, pp. 999-1014 erratum in ibid, 21(1):513–515, 2017 | Zbl

[16] Diestel, Joe; Jarchow, Hans; Tonge, Andrew Absolutely summing operators, Cambridge Studies in Advanced Mathematics, 43, Cambridge University Press, 1995, xvi+474 pages | DOI | MR | Zbl

[17] Dimant, Verónica; Sevilla-Peris, Pablo Summation of Coefficients of Polynomials on $ℓ_{p}$ Spaces, Publ. Mat., Barc., Volume 60 (2016) no. 2, pp. 289-310 | DOI | Zbl

[18] Hardy, Godfrey Harold; Littlewood, John Edensor Bilinear forms bounded in space $[p, q]$ , Quart. J. Math., Volume 5 (1934) no. 1, pp. 241-254 | Zbl

[19] Littlewood, John Edensor On bounded bilinear forms in an infinite number of variables, Quart. J. Math., Volume 1 (1930) no. 1, pp. 164-174 | Zbl

[20] Montanaro, Ashley Some applications of hypercontractive inequalities in quantum information theory, J. Math. Phys., Volume 53 (2012) no. 12, 122206, 15 pages (122206, 15) | DOI | MR | Zbl

[21] Pellegrino, Daniel The optimal constants of the mixed $(ℓ_{1}, ℓ_{2})$ -Littlewood inequality, J. Number Theory, Volume 160 (2016), pp. 11-18 | DOI | MR | Zbl

[22] Pellegrino, Daniel; Santos, Djair; Santos, Joedson Optimal blow up rate for the constants of Khinchin type inequalities, Quaest. Math., Volume 41 (2018) no. 3, pp. 303-318 | DOI

[23] Pellegrino, Daniel; Santos, Joedson; Serrano-Rodríguez, Diana M.; Teixeira, Eduardo V. A regularity principle in sequence spaces and applications, Bull. Sci. Math., Volume 141 (2017) no. 8, pp. 802-837 http://www.sciencedirect.com/science/article/pii/S0007449717300738 | DOI | Zbl

[24] Pellegrino, Daniel; Teixeira, Eduardo V. Towards sharp Bohnenblust–Hille constants, Commun. Contemp. Math., Volume 20 (2018) no. 3, 1750029, 1750029, 33 pages | Zbl

[25] Praciano-Pereira, T. On bounded multilinear forms on a class of $ℓ_{p}$ spaces, J. Math. Anal. Appl., Volume 81 (1981) no. 2, pp. 561-568 | DOI | MR | Zbl

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