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

TL;DR

This paper investigates the boundary behavior of generalized harmonic functions linked with Dunkl operators, introducing a Lusin-type area integral to characterize boundary limits and boundedness in the context of root systems.

Contribution

It provides new characterizations of boundary limits and boundedness of Dunkl harmonic functions using a novel area integral operator.

Findings

Equivalence of boundary limit existence and boundedness conditions

Introduction of a Lusin-type area integral operator for Dunkl harmonic functions

Characterization of boundary behavior in terms of the area integral

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. $x\in E$.