LAUSR

In this paper we define Lipschitz and Triebel–Lizorkin spaces associated with the differential-difference Dunkl operators on $$\mathbb {R}^d$$Rd. We study inclusion relations among them. Next, some interpolation results and continuity results of some important operators (the Dunkl–Poisson semigroup and Dunkl–Flett potentials) on them are established. Also, we prove certain inclusion relations between Dunkl–Sobolev classes $$\mathcal{L}^p_{\alpha ,k}(\mathbb {R}^d)$$Lα,kp(Rd) of positive fractional order $$\alpha $$α, Dunkl–Lipschitz spaces $$\wedge ^k_{\alpha ,p,q}(\mathbb {R}^d)$$∧α,p,qk(Rd) and Dunkl–Triebel–Lizorkin spaces $$F^k_{\alpha ,p,q}(\mathbb {R}^d)$$Fα,p,qk(Rd).