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 $a , b \in ( 0 , \infty )$, the following assertions are equivalent: $\bullet$ There is a constant $C_{ a , b }^{ \mathbb{R} } \geq 1$ such that \[ \left( \sum_{ i = 1 }^\infty \left( \sum_{ j = 1 }^\infty | A( e_i , e_j ) |^a \right)^{ b \times \frac{1}{a} } \right)^{ \frac{1}{b} } \leq C_{ a , b }^{ \mathbb{R} } \| A \| \] for every continuos bilinear form $A \colon c_{0} \times c_{0} \to \mathbb{R}$. $\bullet$ The exponents $a , b$ satisfy $( a , b ) \in [ 1 , \infty ) \times [ 1 , \infty )$ with \[ \frac{1}{a} + \frac{1}{b} \leq \frac{3}{2} . \] Several authors have obtained optimal estimates of the best constant $C_{ a , b }^{ \mathbb{R} }$, for diverse pairs of values $( a , b )$. In this paper we provide the optimal values of $C_{ a , b }^{ \mathbb{R} }$ for all admissible pair of values $( a , b )$. Furthermore, we provide new estimates for $C_{ a , b }^{ \mathbb{C} }$, which are optimal for several pairs of values $( a , b )$. As an application, we prove a variant of Khinchin's inequality for Steinhaus variables.