A Khintchine inequality for central Fourier series on non-Kac compact quantum groups

TL;DR

This paper establishes a Khintchine inequality with operator coefficients for central Fourier series on a broad class of non-Kac compact quantum groups, contrasting classical harmonic analysis limitations.

Contribution

It introduces a novel Khintchine inequality applicable to non-Kac quantum groups, expanding the understanding of Fourier analysis in quantum group theory.

Findings

Khintchine inequality holds for non-Kac quantum groups

Includes examples like $G_q$, $O_F^+$, and $G_{aut}(B,)$

Contrasts classical restrictions on compact Lie groups

Abstract

The study of Khintchin inequalities has a long history in abstract harmonic analysis. While there is almost no possibility of non-trivial Khintchine inequality for central Fourier series on compact connected semisimple Lie groups, we demonstrate a strong contrast within the framework of compact quantum groups. Specifically, we establish a Khintchine inequality with operator coefficients for arbitrary central Fourier series in a large class of non-Kac compact quantum groups. The main examples include the Drinfeld-Jimbo $q$-deformations $G_q$, the free orthogonal quantum groups $O_F^+$, and the quantum automorphism group $G_{aut}(B,\psi)$ with a $\delta$-form $\psi$.