Local boundary behaviour and the area integral of generalized harmonic functions associated with root systems

Abstract

The rational Dunkl operators are commuting differential-reflection operators on the Euclidean space $\RR^d$ associated with a root system. The aim of the paper is to study local boundary behaviour of generalized harmonic functions associated with the Dunkl operators. We introduce a Lusin-type area integral operator $S$ by means of Dunkl's generalized translation and the Dunkl operators. The main results are on characterizations of local existence of non-tangential boundary limits of a generalized harmonic function $u$ in the upper half-space $\RR^{d+1}_+$ associated with the Dunkl operators, and for a subset $E$ of $\RR^d$ invariant under the reflection group generated by the root system, the equivalence of the following three assertions are proved: (i) $u$ has a finite non-tangential limit at $(x,0)$ for a.e. $x\in E$; (ii) $u$ is non-tangentially bounded for a.e. $x\in E$; (iii) $(Su)(x)$ is finite for a.e. $x\in E$.