The sharp constants in the Real Anisotropic Littlewood's $4 / 3$ inequality

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 assertions are equivalent: • There is a constant Ca , bR ≥ 1 such that \[ ( ∑i = 1^∞ ( ∑j = 1^∞ | A( e_i , e_j ) |^a )b × 1/a )1/b ≤ Ca , bR \| A \| \] for every continuos bilinear form A c₀ × c₀ → R. • The exponents $a , b$ satisfy ( a , b ) ∈ [ 1 , ∞ ) × [ 1 , ∞ ) with \[ 1/a + 1/b ≤ 3/2 . \] Several authors have obtained optimal estimates of the best constant Ca , bR, for diverse pairs of values $( a , b )$. In this paper we provide the optimal values of Ca , bR for all admissible pair of values $( a , b )$. Furthermore, we provide new estimates for Ca , bC, which are optimal for several pairs of values $( a , b )$. As an application, we prove a variant of Khinchin's inequality for Steinhaus variables.