Strong Haagerup inequalities on non-Kac free orthogonal quantum groups

Abstract: We present natural analogues of strong Haagerup inequalities on non-Kac free orthogonal quantum groups $O_F^+$ in which $L^p$-analytic problems are harder due to their non-tracial nature. Furthermore, we prove optimality of the inequalities, and apply the obtained results to compute the optimal time for ultracontractivity of the heat semigroup and to distinguish the complex interpolation space $L^p(O_F^+)$ and the real interpolation space $L^{p,p}(O_F^+)$.