Hardy–Littlewood inequalities on compact quantum groups of Kac type

Abstract

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