Noncommutative Khintchine inequalities in interpolation spaces of $L_p$-spaces

TL;DR

This paper extends noncommutative Khintchine inequalities to all interpolation spaces between $L_p$ and $L_2$ for $p<2$, introduces a new deterministic norm equivalence, and discusses limitations and applications in martingale inequalities.

Contribution

It provides the first proof of Khintchine inequalities in $L_{1, obreak ext{,} obreak} ext{infty}$ and introduces a unified approach for all $p$, including $p<1$, with counterexamples in $L_{2, obreak ext{,} obreak} ext{infty}$.

Findings

Khintchine inequalities hold in $L_{1, obreak ext{,} obreak} ext{infty}$.

A new deterministic equivalent for the $RC$-norm is established.

Counterexamples show the limits of existing formulas in $L_{2, obreak ext{,} obreak} ext{infty}$.

Abstract

We prove noncommutative Khintchine inequalities for all interpolation spaces between $L_p$ and $L_2$ with $p<2$. In particular, it follows that Khintchine inequalities hold in $L_{1,\infty}$. Using a similar method, we find a new deterministic equivalent for the $RC$-norm in all interpolation spaces between $L_p$-spaces which unifies the cases $p > 2$ and $p < 2$. It produces a new proof of Khintchine inequalities for $p<1$ for free variables. To complete the picture, we exhibit counter-examples which show that neither of the usual closed formulas for Khintchine inequalities can work in $L_{2,\infty}$. We also give an application to martingale inequalities.