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 : 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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