# Hardy-Littlewood inequalities on compact quantum groups of Kac type

## TL;DR

This paper extends Hardy-Littlewood inequalities to compact quantum groups, establishing explicit $L^p$-$ ext{ell}^p$ bounds and demonstrating their sharpness across various quantum and classical groups.

## Contribution

It introduces a quantum analogue of Hardy-Littlewood inequalities, providing explicit bounds and proving their sharpness for a broad class of quantum groups and classical groups.

## Key findings

- Established $L^p$-$ ext{ell}^p$ inequalities for compact quantum groups.
- Proved the sharpness of these inequalities in several cases.
- Connected inequalities to properties like growth rate and rapid decay of quantum groups.

## Abstract

The Hardy-Littlewood inequality on $\mathbb{T}$ compares the $L^p$-norm of a function with a weighted $\ell^p$-norm of its Fourier coefficients. The approach has recently been studied for compact homogeneous spaces and we study a natural analogue in the framework of compact quantum groups. Especially, in the case of the reduced group $C^*$-algebras and free quantum groups, we establish explicit $L^p-\ell^p$ inequalities through inherent information of underlying quantum group, such as growth rate and rapid decay property. Moreover, we show sharpness of the inequalities in a large class, including $C(G)$ with compact Lie group, $C_r^*(G)$ with polynomially growing discrete group and free quantum groups $O_N^+$, $S_N^+$.