Holomorphic diffusions and boundary behavior of harmonic functions

Abstract

We study a family of differential operators ${L^{\alpha}, \alpha \geq 0}$ in the unit ball $D$ of $C^n$ with $n \geq 2$ that generalize the classical Laplacian, $\alpha = 0$, and the conformal Laplacian, $\alpha = 1/2$ (that is, the Laplace-Beltrami operator for Bergman metric in $D$). Using the diffusion processes associated with these (degenerate) differential operators, the boundary behavior of $L^{\alpha}$-harmonic functions is studied in a unified way for $0 \leq \alpha \leq 1/2$. More specifically, we show that a bounded $L^{\alpha}$-harmonic function in $D$ has boundary limits in approaching regions at almost every boundary point and the boundary approaching region increases from the Stolz cone to the Korányi admissible region as $\alpha$ runs from 0 to 1/2. A local version for this Fatou-type result is also established.