Lipschitz and Triebel--Lizorkin spaces, commutators in Dunkl setting

TL;DR

This paper explores Lipschitz and Triebel--Lizorkin spaces in the Dunkl setting, establishing their relationships, differences from classical spaces, and analyzing commutators of Dunkl operators without Fourier analysis.

Contribution

It introduces new Dunkl-specific Lipschitz and Triebel--Lizorkin spaces, and develops techniques to analyze them without relying on Fourier analysis.

Findings

Dunkl Lipschitz spaces are proper subspaces of classical Lipschitz spaces.

Connections established between Dunkl Lipschitz spaces, Triebel--Lizorkin spaces, and commutators of Dunkl operators.

A discrete weak-type Calderón reproducing formula for Dunkl Triebel--Lizorkin spaces.

Abstract

We first study the Lipschitz spaces $\Lambda_{d}^\beta$ associated with the Dunkl metric, $\beta\in(0,1)$, and prove that it is a proper subspace of the classical Lipschitz spaces $\Lambda^\beta$ on $\mathbb R^N$, as the Dunkl metric and the Euclidean metric are non-equivalent. Next, we further show that the Lipschitz spaces $\Lambda^\beta$ connects to the Triebel--Lizorkin spaces $\dot{ F}^{\alpha,q}_{p,{\rm D}}$ associated with the Dunkl Laplacian $\triangle_{\rm D}$ in $\mathbb R^ N $ and to the commutators of the Dunkl Riesz transform and the fractional Dunkl Laplacian $\triangle_{\rm D}^{-\alpha/2}$, $0<\alpha<\textbf{N}$ (the homogeneous dimension for Dunkl measure), which is represented via the functional calculus of the Dunkl heat semigroup $e^{-t\triangle_{\rm D}}$. The key steps in this paper are a finer decomposition of the underlying space via Dunkl metric and Euclidean metric to bypass the use of Fourier analysis, and a discrete weak-type Calder\'on reproducing formula in these new Triebel--Lizorkin spaces $\dot{ F}^{\alpha,q}_{p,{\rm D}}$.