On the constants of the Bohnenblust-Hille inequality and Hardy--Littlewood inequalities

2014/07/26 by Araujo, Gustavo, Pellegrino, Daniel
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1407.7120

Abstract

In this paper, among other results, we improve the best known estimates for the constants of the generalized Bohnenblust-Hille inequality. These enhancements are then used to improve the best known constants of the Hardy--Littlewood inequality; this inequality asserts that for a positive integer m≥2 with 2m≤ p≤∞ and \mathbbK=ℝ or ℂ there exists a constant C_m,p^\mathbbK≥1 such that, for all continuous m--linear forms T:ℓ_pⁿ×⋯×ℓ_pⁿ→\mathbbK, and all positive integers n,% ( ∑_j₁,...,j_m=1ⁿ\vert T(e_j₁,...,e_j_m% )\vert ^{(2mp)/(mp+p-2m)}) ^{(mp+p-2m)/(2mp)}≤ C_m,p^\mathbbK\Vert T\Vert , and the exponent (2mp)/(mp+p-2m) is sharp. In particular, we show that for p > 2m³-4m²+2m the optimal constants satisfying the above inequality are dominated by the best known estimates for the constants of the m-linear Bohnenblust--Hille inequality. More precisely if γ denotes the Euler--Mascheroni constant, considering the case of complex scalars as an illustration, we show that% \[ C_m,p^ℂ≤∏_j=2^mΓ( 2-\frac1% j) ^(j)/(2-2j)

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