Lipschitz and Triebel–Lizorkin spaces, commutators in Dunkl setting

We first study the Lipschitz spaces Λ_d^β associated with the Dunkl metric, β∈(0,1), and prove that it is a proper subspace of the classical Lipschitz spaces Λ^β on R^N, as the Dunkl metric and the Euclidean metric are non-equivalent. Next, we further show that the Lipschitz spaces Λ^β connects to the Triebel–Lizorkin spaces Ḟ_p,D^α,q associated with the Dunkl Laplacian △_D in R^N and to the commutators of the Dunkl Riesz transform and the fractional Dunkl Laplacian △_D^−α/2, 0<α<N (the homogeneous dimension for Dunkl measure), which is represented via the functional calculus of the Dunkl heat semigroup e^−t△_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 Calderón reproducing formula in these new Triebel–Lizorkin spaces Ḟ_p,D^α,q.