Boundary limits and non-integrability

of -subharmonic functions

in the unit ball of ()

Author:
Manfred Stoll

Journal:
Trans. Amer. Math. Soc. **349** (1997), 3773-3785

MSC (1991):
Primary 31B25, 32F05

MathSciNet review:
1407502

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we consider weighted non-tangential and tangential boundary limits of non-negative functions on the unit ball in that are subharmonic with respect to the Laplace-Beltrami operator on . Since the operator is invariant under the group of holomorphic automorphisms of , functions that are subharmonic with respect to are usually referred to as -subharmonic functions. Our main result is as follows: Let be a non-negative -subharmonic function on satisfying

for some and some , where is the -invariant measure on . Suppose . Then for a.e. ,

uniformly as in each , where for ( when )

We also prove that for the only non-negative -subharmonic function satisfying the above integrability criteria is the zero function.

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Additional Information

**Manfred Stoll**

Affiliation:
Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208

Email:
stoll@math.sc.edu

DOI:
https://doi.org/10.1090/S0002-9947-97-01891-6

Received by editor(s):
May 20, 1995

Received by editor(s) in revised form:
April 1, 1996

Article copyright:
© Copyright 1997
American Mathematical Society