On the upper bounds for the constants of the Hardy–Littlewood inequality

Abstract

Abstract. The best known upper estimates for the constants of the Hardy-Littlewood inequality for m-linear forms on ℓp spaces are of the form √ 2 m−1 . We present better estimates which depend on p and m. An interesting consequence is that if p ≥ m 2 then the constants have a subpolynomial growth as m tends to infinity.

A Gale–Berlekamp permutation-switching problem in higher dimensions

“…for all m-linear forms T : ℓ n p × · · · × ℓ n p → K, all positive integers n. The optimal constants C K m,p , D K m,p are unknown; even the asymptotic behaviour of these constants is unknown. Up to now, the best estimates for C K m,p can be found in [3,4]:…”

Asymptotic Estimates for Unimodular Multilinear Forms with Small Norms on Sequence Spaces

“…The search for the optimal constants involved in Bohnenblust-Hille and Hardy-Littlewood inequalities was efficiently developed in the recent years and the best known constant have polynomial growth (see [10,15,16,18]). Kahane-Salem-Zygmund's inequality turns out to be a formidable (but not complete) tool to provide the optimal exponents on both inequalities (see [1,4,20]). Therefore, a question arises: "How efficient the Kahane-Salem-Zygmund is on providing optimal Bohnenblust-Hille or Hardy-Littlewood constants?…”

Universal Bounds for the Hardy–Littlewood Inequalities on Multilinear Forms

“…The case 0 ≤ 1 p ≤ 1 2 was more explored since it appearance. Several studies have made significant progress in the context 0 ≤ 1 p ≤ 1 2 (see for instance [2,3,5,6,8]). For example, among other results, it was proved in [5,8]…”