Optimal Hardy–Littlewood inequalities uniformly bounded by a universal constant
Annales mathématiques Blaise Pascal, Tome 25 (2018) no. 1, pp. 1-20.

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

ji=11imTej1,,ejmpp-mp-mp2m-12supxi11imT(x1,,xm)

for all continuous m-linear forms T: p ×× p or . 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<pm+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 :
DOI : 10.5802/ambp.371
Classification : 46G25, 47H60
Mots clés : Absolutely summing operators, Hardy–Littlewood inequalities, constants
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.
@article{AMBP_2018__25_1_1_0,
     author = {Albuquerque, Nacib and Ara\'ujo, Gustavo and Maia, Mariana and Nogueira, Tony and Pellegrino, Daniel and Santos, Joedson},
     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},
     volume = {25},
     number = {1},
     year = {2018},
     doi = {10.5802/ambp.371},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/ambp.371/}
}
TY  - JOUR
AU  - Albuquerque, Nacib
AU  - Araújo, Gustavo
AU  - Maia, Mariana
AU  - Nogueira, Tony
AU  - Pellegrino, Daniel
AU  - Santos, Joedson
TI  - Optimal Hardy–Littlewood inequalities uniformly bounded by a universal constant
JO  - Annales mathématiques Blaise Pascal
PY  - 2018
SP  - 1
EP  - 20
VL  - 25
IS  - 1
PB  - Université Clermont Auvergne, Laboratoire de mathématiques Blaise Pascal
UR  - http://archive.numdam.org/articles/10.5802/ambp.371/
DO  - 10.5802/ambp.371
LA  - en
ID  - AMBP_2018__25_1_1_0
ER  - 
%0 Journal Article
%A Albuquerque, Nacib
%A Araújo, Gustavo
%A Maia, Mariana
%A Nogueira, Tony
%A Pellegrino, Daniel
%A Santos, Joedson
%T Optimal Hardy–Littlewood inequalities uniformly bounded by a universal constant
%J Annales mathématiques Blaise Pascal
%D 2018
%P 1-20
%V 25
%N 1
%I Université Clermont Auvergne, Laboratoire de mathématiques Blaise Pascal
%U http://archive.numdam.org/articles/10.5802/ambp.371/
%R 10.5802/ambp.371
%G en
%F AMBP_2018__25_1_1_0
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/

[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 logn 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 1pq, 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 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 ( 1 , 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 p spaces, J. Math. Anal. Appl., Volume 81 (1981) no. 2, pp. 561-568 | DOI | MR | Zbl

Cité par Sources :