On Littlewood–Paley functions associated with the Dunkl operator

A Littlewood–Paley operator associated with the reflection part of the Dunkl operator is introduced and proved to be of type for , based on boundedness of a generalised vector-valued singular integral. This fills a gap for concerning the boundedness of a -function in the Dunkl setting. The paper also supplies new proofs for on the boundedness of various -functions associated with the Dunkl operator.