Lipschitz and Triebel–Lizorkin spaces, commutators in Dunkl setting

Abstract

We first study the Lipschitz spaces Λdβ associated with the Dunkl metric, β∈(0,1), and prove that it is a proper subspace of the classical Lipschitz spaces Λβ on RN, as the Dunkl metric and the Euclidean metric are non-equivalent. Next, we further show that the Lipschitz spaces Λβ connects to the Triebel–Lizorkin spaces Ḟp,Dα,q associated with the Dunkl Laplacian △D in RN 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 Ḟp,Dα,q.