Lipschitz and Triebel-Lizorkin spaces, commutators in Dunkl setting

We first study the Lipschitz spaces Lambda(beta)(d) associated with the Dunkl metric, beta is an element of (0, 1), and prove that it is a proper subspace of the classical Lipschitz spaces Lambda(beta) on 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. F (alpha,q) (p,D) associated with the Dunkl Laplacian Delta(D) in R-N and to the commutators of the Dunkl Riesz transform and the fractional Dunkl Laplacian Delta(-alpha/2)(D) , 0 < alpha < N (the homogeneous dimension for Dunkl measure), which is represented via the functional calculus of the Dunkl heat semigroup e(-t Delta 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 Calderon reproducing formula in these new Triebel-Lizorkin spaces. F (alpha,q) (p,D) . (c) 2023 Elsevier Ltd. All rights reserved.