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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Boundary limits and non-integrability
of $\mathcal {M}$-subharmonic functions
in the unit ball of ${{\mathbb {C}}^{\vphantom {P}}}^{n}$ ($n\ge 1$)


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: In this paper we consider weighted non-tangential and tangential boundary limits of non-negative functions on the unit ball $B$ in ${{\mathbb {C}}^{\vphantom {P}}}^{n}$ that are subharmonic with respect to the Laplace-Beltrami operator $\widetilde {\varDelta }$ on $B$. Since the operator $\widetilde {\varDelta }$ is invariant under the group $\mathcal {M}$ of holomorphic automorphisms of $B$, functions that are subharmonic with respect to $\widetilde {\varDelta }$ are usually referred to as $\mathcal {M}$-subharmonic functions. Our main result is as follows: Let $f$ be a non-negative $\mathcal {M}$-subharmonic function on $B$ satisfying

\begin{equation*}\int _{B} (1-|z|^{2})^{\gamma }f^{p}(z)\,d\lambda (z)< \infty \end{equation*}

for some $p> 0$ and some $\gamma >\min \{n,pn\}$, where $\lambda $ is the $\mathcal {M}$-invariant measure on $B$. Suppose $\tau \ge 1$. Then for a.e. $ \zeta \in S$,

\begin{equation*}f^{p}(z)= o\left ((1-|z|^{2})^{n/\tau -\gamma }\right ) \end{equation*}

uniformly as $z\to \zeta $ in each $\mathcal {T}_{\tau ,\alpha }(\zeta )$, where for $\alpha >0$ ($\alpha >\frac {1}{2}$ when $\tau =1$)

\begin{equation*}\mathcal {T}_{\tau ,\alpha }(\zeta ) = \{z\in B: |1-\langle z,\zeta \rangle |^{\tau } <\alpha (1-|z|^{2}) \}. \end{equation*}

We also prove that for $\gamma \le \min \{n,pn\}$ the only non-negative $\mathcal {M}$-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: http://dx.doi.org/10.1090/S0002-9947-97-01891-6
PII: S 0002-9947(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