The sharp constants in the real anisotropic Littlewood's $4/3$ 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 $A:c_0\times c_0\to \mathbb{R}$. $\bullet$ The values $a,b$ satisfy $a,b\ge 1$ and $\frac{1}{a}+\frac{1}{b}\le \frac{3}{2}$. 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 ${\mathsf{L}}_{1,q}^{\mathbb{R}}$.

Final version