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 T compares the Lp-norm of a function with a weighted 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 Lp 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, Cr(G) with G a polynomially growing discrete group and free quantum groups ON+, SN+.

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