Weak factorizations for Hardy spaces in the Dunkl setting

In this paper, we establish the weak factorizations of the Hardy space associated with the Dunkl operator via the bilinear forms of Dunkl–Riesz transforms \(\{{\mathcal {R}}_{j}\}_{j=1}^{d}.\) Note that the kernels of \(\{{\mathcal {R}}_{j}\}_{j=1}^{d}\) involve both the Euclidean and the Dunkl metrics, which are not equivalent. As an application, we provide a new proof for the sufficiency of characterization of the \({\textrm{BMO}}\) space associated to the Dunkl operator via the commutators of \(\{{\mathcal {R}}_{j}\}_{j=1}^{d}.\)