The strong Haagerup inequality for q-circular systems

Together with Speicher, in 2007 the first author proved the strong Haagerup inequality for operator norms of homogeneous holomorphic polynomials in freely independent $\mathscr{R}$-diagonal elements (including in particular circular random variables); the inequality improved the bound from the original Haagerup inequality to grow with $\sqrt{n}$, rather than linearly in $n$, on homogeneous polynomials of degree $n$. In this paper, we prove a similar inequality for $q$-circular systems for $|q|<1$, generalizing the free case when $q=0$. In particular, we prove the strong Haagerup inequality for systems exhibiting neither free independence nor $\mathscr{R}$-diagonality. As an application, we prove a strong ultracontractivity theorem for the $q$-Ornstein--Uhlenbeck semigroup, and prove sharp rates for the Haagerup and ultracontractive inequalities.