Triebel-Lizorkin spaces in Dunkl setting

TL;DR

This paper develops Triebel-Lizorkin function spaces within the Dunkl setting, leveraging group-induced metrics and singular integral operators to extend harmonic analysis tools to reflection group contexts.

Contribution

It introduces Triebel-Lizorkin spaces in the Dunkl setting, utilizing new Calderon reproducing formulas, test functions, and wavelet-type decompositions for the first time.

Findings

Established Triebel-Lizorkin spaces in Dunkl setting

Developed new Calderon reproducing formula in L^2

Defined atomic decomposition for Dunkl Hardy spaces

Abstract

We establish Triebel-Lizorkin spaces in the Dunkl setting which are associated with finite reflection groups on the Euclidean space. The group structures induce two nonequivalent metrics: the Euclidean metric and the Dunkl metric. In this paper, the L^2 space and the Dunkl-Calderon-Zygmund singular integral operator in the Dunkl setting play a fundamental role. The main tools used in this paper are as follows: (i) the Dunkl-Calderon-Zygmund singular integral operator and a new Calderon reproducing formula in L^2 with the Triebel-Lizorkin space norms; (ii) new test functions in terms of the L^2 functions and distributions; (iii) the Triebel-Lizorkin spaces in the Dunkl setting which are defined by the wavelet-type decomposition with norms and the analogous atomic decomposition of the Hardy spaces.