Boundary Values of Generalized Harmonic Functions Associated with the Rank-One Dunkl Operator

Abstract

We consider the local boundary values of generalized harmonic functions associated with the rank-one Dunkl operator $D$ in the upper half-plane $R^{2}_+=R\times(0,\infty)$, where
                           $$(Df)(x)=f'(x)+(\lambda/x)[f(x)-f(-x)]$$
for given $\lambda\ge0$. A $C^2$ function $u$ in $R^{2}_+$ is said to be $\lambda$-harmonic if $(D_x^2+\partial_{y}^2)u=0$. For a $\lambda$-harmonic function $u$ in $R^{2}_+$ and for a subset $E$ of $\partial R^{2}_+=R$ symmetric about $y$-axis, we prove that the following three assertions are equivalent: (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) $(Su)(x)<\infty$ for a.e. $x\in E$, where $S$ is a Lusin-type area integral associated with the Dunkl operator $D$.