Twisted Fourier transforms on non-Kac compact quantum groups
Sang-Gyun Youn
TL;DR
This work introduces an analytic family of twisted Fourier transforms $\{\mathcal{F}^{(x)}_p\}$ on non-Kac compact quantum groups and proves a sharp Hausdorff–Young inequality for $1\le p<2$ in the range $0\le x\le 1$. By developing a duality framework, it derives a strengthened twisted rapid decay property for discrete quantum groups with polynomial growth, including duals of Drinfeld–Jimbo $q$-deformations. The paper then characterizes the boundedness set $I(\mathbb{G},p)$, showing $[0,1]$ is guaranteed under polynomial (or sub-exponential) growth, while in some non-coamenable cases like $O_F^+$ the interval can extend beyond $[0,1]$. These results illuminate how non-tracial (non-Kac) structures influence harmonic analysis and connect growth conditions to sharp analytic inequalities in the quantum group setting.
Abstract
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 $0\leq x \leq 1$. As an application, 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. 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.