Harmonic analysis in Dunkl settings

Let L be the Dunkl Laplacian on the Euclidean space R^N associated with a normalized root R and a multiplicity function k(ν) ≥ 0, ν ∈ R. In this paper, we first prove that the Besov and Triebel-Lizorkin spaces associated with the Dunkl Laplacian L are identical to the Besov and Triebel-Lizorkin spaces defined in the space of homogeneous type (R^N,‖⋅‖, dw), where dw(x) = ∏_ν∈R〈ν,x〉^k(ν)dx. Next, consider the Dunkl transform denoted by F. We introduce the multiplier operator T_m, defined as T_mf = F⁻¹(mFf), where m is a bounded function defined on R^N. Our second aim is to prove multiplier theorems, including the Hörmander multiplier theorem, for T_m on the Besov and Tribel-Lizorkin spaces in the space of homogeneous type (R^N,‖⋅‖, dw). Importantly, our findings present novel results, even in the specific case of the Hardy spaces.