The sharp constants in the real anisotropic Littlewood's $\boldsymbol{4 / 3}$ inequality and applications

Abstract

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 conditions are equivalent: $\bullet$ There is an optimal constant $\mathsf{L}_{ a , b }^{ \mathbb{R} } \in [ 1 , \infty )$ such that \[ \Biggl ( \, \sum_{ k = 1 }^{ \infty } \biggl ( \, \sum_{ j = 1 }^{ \infty } \bigl \lvert A \bigl ( \boldsymbol{e}^{ (k) } , \boldsymbol{e}^{ (j) } \bigr ) \bigr \rvert^a \biggr )^{ \frac{b}{a} } \Biggr )^{ \frac{1}{b} } \leq \mathsf{L}_{ a , b }^{ \mathbb{R} } \cdot \lVert A \rVert \] for every continuous bilinear form $A \colon c_0 \times c_0 \to \mathbb{R}$. $\bullet$ The values $a , b$ satisfy $a , b \geq 1$ and $\frac{1}{a} + \frac{1}{b} \leq \frac{3}{2}$. Several authors have obtained the values of $\mathsf{L}_{ a , b }^{ \mathbb{R} }$ for diverse pairs $( a , b )$. In this paper we provide the complete list of such optimal values, as well as new estimates for $\mathsf{L}_{ a , b }^{ \mathbb{C} }$ (the analog for continuous $\mathbb{C}$-bilinear forms), which are exact in several cases. As an application we prove, in terms of the values $\mathsf{L}_{ 1 , r }^{ \mathbb{C} }$, a variant of Khinchin's inequality for Steinhaus variables, and we provide estimates for the optimal $( q , s )$-cotype constants of the spaces $\ell_1 ( \mathbb{K} )$ (with $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$) in terms of the values $\mathsf{L}_{ 1 , q }^{ \mathbb{R} }$.