Optimal Hardy–Littlewood inequalities uniformly bounded by a universal constant

摘要

The m -linear version of the Hardy–Littlewood inequality for m -linear forms on ? p spaces and m p 2 m , recently proved by Dimant and Sevilla-Peris, asserts that ∑ j i = 1 1 ≤ i ≤ m ∞ T e j 1 , ? , e j m p p - m p - m p ≤ 2 m - 1 2 sup x i ≤ 1 1 ≤ i ≤ m T ( x 1 , ? , x m ) 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 p ≤ 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.