Optimal constants for a mixed Littlewood type inequality

2016/04/21 by Nogueira, Tony, Núñez-Alarcón, Daniel, Pellegrino, Daniel
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1604.06323

Abstract

For p∈\lbrack2,∞] a mixed Littlewood-type inequality asserts that there is a constant C_(m),p≥1 such that ( ∑_i₁=1^∞( ∑_i₂,...,i_m=1^∞ |T(e_i₁,...,e_i_m)|²) ^{(1)/(2)(p)/(p-1)}) ^(p-1)/(p)≤ C_(m),p\Vert T\Vert for all continuous real-valued m-linear forms on ℓ_p× c₀ ×…× c₀ (when p=∞, ℓ_p is replaced by c₀). We prove that for p>2.18006 the optimal constants C_(m),p are ( 2^{(1)/(2)-(1)/(p)}) ^m-1. When p=∞, we recover the best constants of the mixed ( ℓ₁,ℓ₂) -Littlewood inequality.

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