Maximal Operator, Cotlar’s Inequality and Pointwise Convergence for Singular Integral Operators in Dunkl Setting

Abstract

We establish the maximal operator, Cotlar’s inequality and pointwise convergence in the Dunkl setting for the (nonconvolution type) Dunkl–Calderón–Zygmund operators introduced recently in Tan et al. (https://arxiv.org/abs/2204.01886). The fundamental geometry of the Dunkl setting involves two nonequivalent metrics: the Euclidean metric and the Dunkl metric deduced by finite reflection groups, and hence the classical methods do not apply directly. The key idea is to introduce truncated singular integrals and the maximal singular integrals by the Dunkl metric and the Euclidean metric. We show that these two kind of truncated singular integrals are dominated by the Hardy–Littlewood maximal function, which yields the Cotlar’s inequalities and hence the boundedness of maximal Dunkl–Calderón–Zygmund operators. Further, as applications, two equivalent pointwise convergences for Dunkl–Calderón–Zygmund operators are obtained.