$q$-Chaos

TL;DR

This paper investigates $L_p$ norm estimates for homogeneous polynomials of $q$-Gaussian variables, revealing that estimates vary significantly with $q$, especially at extremal values, and are similar to free cases for certain ranges.

Contribution

It provides new $L_p$ estimates for $q$-Gaussian polynomials across the entire range of $q$, highlighting differences at extremal points and their relation to free probability.

Findings

$L_p$ estimates for $-1<q<1$ are similar to free case for $1 \\leq p \\leq 2$

Estimates for $2 \\leq p \\leq \\infty$ depend strongly on $q$

Extremal cases $q= \\pm 1$ yield different formulas.

Abstract

We consider the $L_p$ norm estimates for homogeneous polynomials of $q$-gaussian variables ($-1\leq q\leq 1$). When $-1<q<1$ the $L_p$ estimates for $1\leq p \leq 2$ are essentially the same as the free case ($q=0$), whilst the $L_p$ estimates for $2\leq p \leq \infty$ show a strong $q$-dependence. Moreover, the extremal cases $q = \pm 1$ produce decisively different formulae.