Universal bounds for the Hardy--Littlewood inequalities on multilinear\n forms

2018/03/04 by Gustavo Neves de Araújo, Araújo, Gustavo, Kleber Câmara +1
Mathematics · #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #Advanced Operator Algebra Research #FOS: Mathematics #Functional Analysis (math.FA)

paper · pdf · doi:10.48550/arxiv.1803.01397

openalex publication_date 2018/03/04 · openalex created_date 2022/10/06 · openalex updated_date 2026/07/28

Abstract

The Hardy--Littlewood inequalities for multilinear forms on sequence spaces\nstate that for all positive integers m,n\≥2 and all m-linear forms\nT:\ℓ_p₁ⁿ\×\⋯\×\ℓ_p_mⁿ\→ mathbbK\n( mathbbK=\ℝ or \ℂ) there are constants C_m\≥1 (not\ndepending on n) such that \
left(
sum_j₁,
ldots,j_m=1ⁿ
left
vert\nT(e_j₁,
ldots,e_j_m)
right
vert
^rho
right) ^
frac1
rho
leq\nC_m
sup_
left
Vert x₁
right
Vert ,
dots,
left
Vert x_m
right
Vert
leq\n1
left
vert T(x₁,
dots,x_m)
right
vert, where\n\ρ= frac2mm+1-2\( frac1p₁+\⋯+ frac1p_m\) if\n0\≤ frac1p₁+\⋯+ frac1p_m\≤\(1)/(2) or\n\ρ= frac11-\( frac1p₁+\⋯+ frac1p_m\) if\n\(1)/(2)\≤ frac1p₁+\⋯+ frac1p_m<1. Good estimates for\nthe Hardy-Littlewood constants are, in general, associated to applications in\nMathematics and even in Physics, but the exact behavior of these constants is\nstill unknown. In this note we give some new contributions to the behavior of\nthe constants in the case\n\(1)/(2)\≤ frac1p₁+\⋯+ frac1p_m<1. As a consequence of\nour main result, we present a generalization and a simplified proof of a result\ndue to Aron et al. on certain Hardy--Littlewood type inequalities.\n

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