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

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

 ${\left(\sum _{\begin{array}{c}{j}_{i}=1\\ 1\le i\le m\end{array}}^{\infty }{\left|T\left({e}_{{j}_{1}},\cdots ,{e}_{{j}_{m}}\right)\right|}^{\frac{p}{p-m}}\right)}^{\frac{p-m}{p}}\le {2}^{\frac{m-1}{2}}\underset{\begin{array}{c}∥{x}_{i}∥\le 1\\ 1\le i\le m\end{array}}{sup}\left|T\left({x}_{1},\cdots ,{x}_{m}\right)\right|$

for all continuous $m$-linear forms $T:{\ell }_{p}×\cdots ×{\ell }_{p}\to ℝ$ 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 ${2}^{\left(m-1\right)/2}$ replaced by ${2}^{\left(m-1\right)\left(p-m\right)/p}$. In particular, we conclude that for $m 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.

Published online:
DOI: 10.5802/ambp.371
Classification: 46G25,  47H60
Keywords: Absolutely summing operators, Hardy–Littlewood inequalities, constants
Nacib Albuquerque 1; Gustavo Araújo 2; Mariana Maia 3, 1; Tony Nogueira 4, 1; Daniel Pellegrino 1; Joedson Santos 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.
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Nacib Albuquerque; Gustavo Araújo; Mariana Maia; Tony Nogueira; Daniel Pellegrino; Joedson Santos. Optimal Hardy–Littlewood inequalities uniformly bounded by a universal constant. Annales mathématiques Blaise Pascal, Volume 25 (2018) no. 1, pp. 1-20. doi : 10.5802/ambp.371. https://ambp.centre-mersenne.org/articles/10.5802/ambp.371/

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