On the Sobolev embedding properties for compact matrix quantum groups of Kac type

Sang-Gyun Youn

doi:10.3934/cpaa.2020148

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2020, Volume 19, Issue 6: 3341-3366. Doi: 10.3934/cpaa.2020148

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On the Sobolev embedding properties for compact matrix quantum groups of Kac type

Sang-Gyun Youn

Department of Mathematics Education, Seoul National University, Gwanak-ro 1, Gwanak-gu, Seoul, 08826, Republic of Korea

Received: August 31, 2019

Revised: November 30, 2019

Published: March 2020

This research was supported by the Natural Sciences and Engineering Research Council of Canada and by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1C1C1A01009681)

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We study the optimal order of natural analogues of Sobolev embedding properties within the framework of compact matrix quantum groups of Kac type. One of the main results of this paper is that the optimal order is given by the polynomial growth order of dual discrete quantum groups in a broad class, which covers all connected compact Lie groups, duals of polynomially growing discrete groups, $ O_2^+ $ and $ S_4^+ $. Outside the realm of co-amenable compact quantum groups, we prove that the optimal order is $ 3 $ for duals of free groups and free quantum groups $ O_N^+ $ and $ S_N^+ $, and that Sobolev embedding properties can be generalized for all compact matrix quantum groups of Kac type whose duals have the rapid decay property. In addition, we generalize sharpened Hausdorff-Young inequalities, compute degrees of the rapid decay property for duals of $ O_N^+,S_N^+ $ and prove sharpness of Hardy-Littlewood inequalities on duals of free groups.

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Mathematics Subject Classification: Primary 43A15, 46L52, Secondary 20G42, 81R15.

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