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 ${\ell }_{p}$ 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.

Publié le : 2018-07-01
DOI : https://doi.org/10.5802/ambp.371
Classification : 46G25,  47H60
Mots clés: Absolutely summing operators, Hardy–Littlewood inequalities, constants
@article{AMBP_2018__25_1_1_0,
author = {Nacib Albuquerque and Gustavo Ara\'ujo and Mariana Maia and Tony Nogueira and Daniel Pellegrino and Joedson Santos},
title = {Optimal Hardy--Littlewood inequalities uniformly bounded by a universal constant},
journal = {Annales Math\'ematiques Blaise Pascal},
publisher = {Universit\'e Clermont Auvergne, Laboratoire de math\'ematiques Blaise Pascal},
volume = {25},
number = {1},
year = {2018},
pages = {1-20},
doi = {10.5802/ambp.371},
language = {en},
url = {ambp.centre-mersenne.org/item/AMBP_2018__25_1_1_0/}
}
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, Tome 25 (2018) no. 1, pp. 1-20. doi : 10.5802/ambp.371. https://ambp.centre-mersenne.org/item/AMBP_2018__25_1_1_0/

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