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Holomorphic diffusions and boundary behavior of harmonic functions. (English) Zbl 0891.60072

Summary: We study a family of differential operators \(\{L^\alpha,\;\alpha\geq 0\}\) in the unit ball \(D\) of \(\mathbb{C}^n\) 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.

MSC:

60J45 Probabilistic potential theory
31B25 Boundary behavior of harmonic functions in higher dimensions
60J60 Diffusion processes
31B10 Integral representations, integral operators, integral equations methods in higher dimensions
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