Finite Decomposition of Herz-Type Hardy Spaces for the Dunkl Operator

The corresponding Herz-type Hardy spaces to new weighted Herz spaces \(HK^{\beta ,p}_{\alpha ,q}\) associated with the Dunkl operator on \(\mathbb {R}\) have been characterized by atomic decompositions. Later a new characterization of \(HK^{\beta ,p}_{\alpha ,q}\) on the real line is introduced. This helped us in the work to characterize that the norms of the Herz-type Hardy spaces for the Dunkl Operator can be achieved by finite central atomic decomposition in some dense subspaces of them. Secondly, as an application we prove that a sublinear operator satisfying many conditions can be uniquely extended to a bounded operator in the Herz-type Hardy spaces for the Dunkl Operator.