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://www.numdam.org/articles/10.5802/ambp.371/

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