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

We first study the Lipschitz spaces ${\mathrm{\Lambda }}_d^{\beta }$ associated with the Dunkl metric, $\beta \in (0,1)$, and prove that it is a proper subspace of the classical Lipschitz spaces ${\mathrm{\Lambda }}^{\beta }$ on ${\mathbb{R}}^N$, as the Dunkl metric and the Euclidean metric are non-equivalent. Next, we further show that the Lipschitz spaces ${\mathrm{\Lambda }}^{\beta }$ connects to the Triebel–Lizorkin spaces ${\dot{F}}_{p,\mathrm{D}}^{\alpha ,q}$ associated with the Dunkl Laplacian ${\mathrm{△}}_{\mathrm{D}}$ in ${\mathbb{R}}^N$ and to the commutators of the Dunkl Riesz transform and the fractional Dunkl Laplacian ${\mathrm{△}}_{\mathrm{D}}^{-\alpha \mathrm{/}2}$, $0<\alpha <\mathbf{\text{N}}$ (the homogeneous dimension for Dunkl measure), which is represented via the functional calculus of the Dunkl heat semigroup $e^{-t{\mathrm{△}}_{\mathrm{D}}}$. 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 ${\dot{F}}_{p,\mathrm{D}}^{\alpha ,q}$.