Hardy-Littlewood inequalities on compact quantum groups of Kac type

The Hardy-Littlewood inequality on $\mathbb{T}$ compares the $L^p$-norm of a function with a weighted ${\mathrm{\ell }}^p$-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 $C^{\ast }$-algebras and free quantum groups, we establish explicit $L^p-{\mathrm{\ell }}^p$ 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, $C_r^{\ast }(G)$ with polynomially growing discrete group and free quantum groups $O_N^{+}$, $S_N^{+}$.