Nontangential Limits of Poisson Integrals Associated to Dunkl Operators for Dihedral Groups

Abstract

In this paper we study the differentiation and maximal functions of complex Borel measures on the unit circle of $\mathbb{C}$ with respect to the measures associated to Dunkl differential-difference operators for dihedral groups. We prove that the Poisson integrals corresponding to these differential-difference operators have nontangential limits almost everywhere. Our approach relies on the proof of the doubling condition to obtain an appropriate covering lemma.