Abstract
We present natural analogues of strong Haagerup inequalities on non-Kac free orthogonal quantum groups OF+ in which Lp-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 Lp(OF+) and the real interpolation space Lp,p(OF+).
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