Vol. 11, No. 1, 2018

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Hardy–Littlewood inequalities on compact quantum groups of Kac type

Sang-Gyun Youn

Vol. 11 (2018), No. 1, 237–261
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 ${O}_{N}^{+}$, ${S}_{N}^{+}$.

Keywords
Hardy–Littlewood inequality, quantum groups, Fourier analysis
Mathematical Subject Classification 2010
Primary: 20G42, 43A15, 46L51, 46L52