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 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.

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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