Twisted Fourier transforms on non-Kac compact quantum groups

We introduce an analytic family of twisted Fourier transforms $\left\{\mathcal{F}^{(x)}_p\right\}_{x\in \mathbb{R},p\in [1,2)}$ for non-Kac compact quantum groups and establish a sharpened form of the Hausdorff-Young inequality in the range $0\leq x \leq 1$. Furthermore, we prove that the range $0\leq x \leq 1$ is both necessary and sufficient for the boundedness of $\mathcal{F}^{(x)}_p$ under the assumption of sub-exponential growth on the dual discrete quantum group. We also show that the range of boundedness of $\mathcal{F}^{(x)}_p$ can be strictly extended beyond $[0,1]$ for certain non-Kac and non-coamenable free orthogonal quantum groups. As applications, we derive a stronger form of the twisted rapid decay property for polynomially growing non-Kac discrete quantum groups, including the duals of the Drinfeld-Jimbo $q$-deformations, and construct an explicit contractive, but non-completely bounded, representation of the convolution algebra of any non-Kac free orthogonal quantum group.