Holomorphic diffusions and boundary behavior of harmonic functions


Journal Article

We study a family of differential operators {Lα, α ≥ 0} in the unit ball D of Cn with n ≥ 2 that generalize the classical Laplacian, α = 0, and the conformal Laplacian, α = 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 Lα-harmonic functions is studied in a unified way for 0 ≤ α ≤ 1/2. More specifically, we show that a bounded Lα-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 α runs from 0 to 1/2. A local version for this Fatou-type result is also established.

Full Text

Duke Authors

Cited Authors

  • Chen, ZQ; Durrett, R; Ma, G

Published Date

  • January 1, 1997

Published In

Volume / Issue

  • 25 / 3

Start / End Page

  • 1103 - 1134

International Standard Serial Number (ISSN)

  • 0091-1798

Digital Object Identifier (DOI)

  • 10.1214/aop/1024404507

Citation Source

  • Scopus