Fourier multipliers for Triebel-Lizorkin spaces on compact Lie groups

TL;DR

This paper establishes criteria for the boundedness of Fourier multipliers on Triebel-Lizorkin spaces over compact Lie groups, extending classical results like the H"ormander-Mihlin theorem to a broader functional setting.

Contribution

It provides new boundedness criteria for Fourier multipliers on Triebel-Lizorkin spaces on compact Lie groups, utilizing the difference structure of the unitary dual.

Findings

Criteria based on H"ormander-Mihlin-Marcinkiewicz condition

Extension of classical Fourier multiplier theorems to Triebel-Lizorkin spaces

Application to the sharp H"ormander-Mihlin theorem on Lebesgue spaces

Abstract

We investigate the boundedness of Fourier multipliers on a compact Lie group when acting on Triebel-Lizorkin spaces. Criteria are given in terms of the H\"ormander-Mihlin-Marcinkiewicz condition. In our analysis, we use the difference structure of the unitary dual of a compact Lie group. Our results cover the sharp H\"ormander-Mihlin theorem on Lebesgue spaces and also other historical results on the subject.