A self-dual Dunkl wavelet frame and Besov--Triebel--Lizorkin spaces

We construct a self-dual Parseval wavelet frame associated with the Dunkl transform and use it to characterize the corresponding Besov and Triebel-Lizorkin spaces for arbitrary smoothness and the full range of admissible indices. We also establish differential atomic decompositions with atoms supported on dilated Euclidean dyadic cubes, satisfying ordinary derivative estimates and polynomial moment indices. A key difficulty is the interaction between Euclidean dyadic cubes and the orbit geometry associated with the Dunkl operators. As applications, we obtain fractional chain rules and a Jackson inequality for nonlinear wavelet approximation in $L^p(dw)$.