$q$-Chaos

Abstract

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

22 pages