Hardy–Littlewood inequalities on compact quantum groups of Kac type

Youn, Sang-Gyun

doi:10.2140/apde.2018.11.237

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Hardy–Littlewood inequalities on compact quantum groups of Kac type

Sang-Gyun Youn

Vol. 11 (2018), No. 1, 237–261

DOI: 10.2140/apde.2018.11.237

Abstract

The Hardy–Littlewood inequality on the circle group $T$ compares the $L^{p}$ -norm of a function with a weighted $ℓ^{p}$ -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 $C^{*}$ -algebras and free quantum groups, we establish explicit $L^{p} - ℓ^{p}$ 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, $C_{r}^{*} (G)$ 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

Milestones

Received: 16 February 2017

Revised: 16 June 2017

Accepted: 24 July 2017

Published: 17 September 2017

Authors

Sang-Gyun Youn
Department of Mathematical Sciences Seoul National University Seoul South Korea

Vol. 11, No. 1, 2018