2024/07/09 by Caro-Montoya, Nicolás, Núñez-Alarcón, Daniel, Serrano-Rodríguez, Diana
#11Y60 #42B08 #46B09 #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2407.06804
The real anisotropic Littlewood's 4 / 3 inequality is an extension of a famous result obtained in 1930 by J. E. Littlewood. It asserts that, for a , b ∈ ( 0 , ∞ ), the following conditions are equivalent: \bullet There is an optimal constant L a , b ℝ ∈ [ 1 , ∞ ) such that \Biggl ( ∑ k = 1 ∞ \biggl ( ∑ j = 1 ∞ | A ( \boldsymbole (k) , \boldsymbole (j) ) |a \biggr ) (b)/(a) \Biggr ) (1)/(b) ≤ L a , b ℝ ⋅ ‖ A ‖ for every continuous bilinear form A \colon c0 × c0 → ℝ. \bullet The values a , b satisfy a , b ≥ 1 and (1)/(a) + (1)/(b) ≤ (3)/(2). Several authors have obtained the values of L a , b ℝ for diverse pairs ( a , b ). In this paper we provide the complete list of such optimal values, as well as new estimates for L a , b ℂ (the analog for continuous ℂ-bilinear forms), which are exact in several cases. As an application we prove, in terms of the values L 1 , r ℂ , a variant of Khinchin's inequality for Steinhaus variables, and we provide estimates for the optimal ( q , s )-cotype constants of the spaces ℓ1 ( \mathbbK ) (with \mathbbK = ℝ or ℂ) in terms of the values L 1 , q ℝ .