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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References:

 [1] Bass, R. F. (1995). Probabilistic Techniques in Analysis. Springer, New York. · Zbl 0817.60001 [2] Debiard, A. (1979). Comparison des espaces Hp géométrique et probabilists au-dessus de l’espace Hermitien hyperbolique. Bull. Sci. Math. (2) 103 305-351. · Zbl 0412.31007 [3] Durrett, R. (1984). Brownian Motion and Martingales in Analysis. Wadsworth, Belmont, CA. · Zbl 0554.60075 [4] Fukushima, M. and Okada, M. (1987). On Dirichlet forms for plurisubharmonic functions. Acta Math. 159 171-213. · Zbl 0637.32013 [5] Gilbarg, D. and Trudinger, N. S. (1983). Elliptic Partial Differential Equations of Second Order, 2nd ed. Springer, New York. · Zbl 0562.35001 [6] Hakim, M. and Sibony, N. (1983). Fonctions holomophes bornées et limites tangentielles. Duke Math. J. 50 133-141. · Zbl 0514.32003 [7] Hua, L. K. (1969). Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains. Amer. Math. Soc., Providence, RI. · Zbl 0507.32025 [8] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Springer, New York. · Zbl 0734.60060 [9] Korányi, A. (1969). Harmonic functions on Hermitian hyperbolic space. Trans. Amer. Math. Soc. 135 507-516. · Zbl 0174.38801 [10] Krantz, S. (1991). Invariant metrics and the boundary behavior of holomorphic functions in domains in Cn. Journal of Geometric Analysis 1 71-97. · Zbl 0728.32002 [11] Rudin, W. (1980). Function Theory in the Unit Ball of Cn. Springer, New York. · Zbl 0495.32001 [12] Stein, E. M. (1972). Boundary Behavior of Holomorphic Functions of Several Complex Variables. Princeton Univ. Press. · Zbl 0242.32005
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