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Transactions of the American Mathematical Society

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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
DOI: https://doi.org/10.1090/S0002-9947-97-01891-6
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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  • [BC] R. D. Berman and W. S. Cohn, Littlewood theorems for limits and growth of potentials along level sets of Hölder continuous functions, Amer. J. of Math. 114 (1991), 185-227. MR 93c:31006
  • [CA] G. T. Cargo, Angular and tangential limits of Blaschke products and their successive derivatives, Canad. J. Math. 14 (1962), 334-348. MR 25:204
  • [CO] W. S. Cohn, Non-isotropic Hausdorff measure and exceptional sets for holomorphic Sobolev functions, Illinois J. Math. 33 (1989), 673-690.MR 90j:32008
  • [FS] C. Fefferman and S. Stein, $H^{p}$ spaces of several variables, Acta Math 129 (1972), 137-193. MR 56:6263
  • [GE] F. W. Gehring, On the radial order of subharmonic functions, Jour. Math. Soc. Japan 9 (1957), 77-79. MR 19:131e
  • [HSY] K. T. Hahn, M. Stoll, and H. Youssfi, Invariant potentials and tangential boundary behavior of $\mathcal {M}$-subharmonic functions in the unit ball, Complex Variables 28 (1995), 67-96.
  • [HY] K. T. Hahn and H. Youssfi, Tangential boundary behavior of $\mathcal {M}-$harmonic Besov functions in the unit ball, J. Math. Analysis and Appl. 175 (1993), 206-221. MR 94c:32006
  • [HL] G. H. Hardy and J. E. Littlewood, Some properties of conjugate functions, J. Reine Angew. Math. 167 (1932), 403-423.
  • [KI1] J. R. Kinney, Boundary behavior of Blaschke products in the unit circle, Proc. Amer. Math. Soc. 12 (1961), 484-488. MR 23:A2533
  • [KI2] -, Tangential limits of functions of the class $S_{\alpha }$, Proc. Amer. Math. Soc. 14 (1963), 68-70. MR 26:1466
  • [KO] P. Koosis, Introduction to $H^{p}$ Spaces, London Math. Soc. Lecture Notes 40 (1980). MR 81c:30062
  • [MA] B. D. MacCluer, Compact composition operators on $H^{p}(B_{n})$, Michigan Math. J. 32 (1985), 237-248. MR 86g:47037
  • [NRS] A. Nagel, W. Rudin, and J. H. Shapiro, Tangential boundary behavior of functions in Dirichlet-type spaces, Annals of Math. 116 (1982), 331-360. MR 84a:31002
  • [PA] M. Pavlovic, Inequalities for the gradient of eigenfunctions of the invariant laplacian in the unit ball, Indag. Mathem., N.S. 2 (1991), 89-98. MR 92d:32008
  • [RU] W. Rudin, Function Theory in the Unit Ball of $\mathbb {C}^{n}$, Springer-Verlag, New York, 1980. MR 82i:32002
  • [RZ] L. Rzepecki, Boundary behavior of non-isotropic potentials in the unit ball of ${{\mathbb C}^{\vphantom {P}}}^n$, Ph. D. Dissertation, University of South Carolina (1995).
  • [ST1] M. Stoll, Tangential boundary limits of invariant potentials in the unit ball of $\mathbb {C}^{n}$, J. Math. Analysis & Appl. 177 (1993), 553-571. MR 94h:32020
  • [ST2] -, Invariant Potential Theory in the Unit Ball of $\mathbb {C}^{n}$, London Math. Soc. Lect. Note Series 199 (1994). MR 96f:31011
  • [ST3] -, Non-isotropic Hausdorff capacity of exceptional sets of invariant potentials, Potential Analysis 4 (1995), 141-155. MR 96b:31011
  • [SU] J. Sueiro, Tangential boundary limits and exceptional sets for holomorphic functions in Dirichlet-type spaces, Math. Ann. 286 (1990), 661-678. MR 91b:32008
  • [SZ] N. Suzuki, Nonintegrability of harmonic functions in a domain, Japan J. Math. 16 (1990), 269-278. MR 91m:31003
  • [TS] M. Tsuji, Potential Theory in Modern Function Theory, Chelsea Publ. Co., New York, N.Y., 1975. MR 54:2990
  • [TW] J. B. Twomey, Tangential limits for certain classes of analytic functions, Mathematika 36 (1989), 39-49. MR 91b:30100
  • [UL] D. Ullrich, Radial limits of $\mathcal {M}$-subharmonic functions, Trans. Amer. Math. Soc. 292 (1985), 501-518. MR 87a:31007
  • [ZH] S. Zhao, On the weighted $L^{p}$-integrability of nonnegative $\mathcal {M}$-superharmonic functions, Proc. Amer. Math. Soc. 115 (3) (1992), 677-685. MR 92i:31005

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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

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