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.
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     title = {Optimal {Hardy{\textendash}Littlewood} inequalities uniformly bounded by a universal constant},
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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://www.numdam.org/articles/10.5802/ambp.371/

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