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

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

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.

**[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,*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 -subharmonic functions in the unit ball*, Complex Variables**28**(1995), 67-96.**[HY]**K. T. Hahn and H. Youssfi,*Tangential boundary behavior of 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*, Proc. Amer. Math. Soc.**14**(1963), 68-70. MR**26:1466****[KO]**P. Koosis,*Introduction to Spaces*, London Math. Soc. Lecture Notes**40**(1980). MR**81c:30062****[MA]**B. D. MacCluer,*Compact composition operators on*, 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*, Springer-Verlag, New York, 1980. MR**82i:32002****[RZ]**L. Rzepecki,*Boundary behavior of non-isotropic potentials in the unit ball of*, Ph. D. Dissertation, University of South Carolina (1995).**[ST1]**M. Stoll,*Tangential boundary limits of invariant potentials in the unit ball of*, J. Math. Analysis & Appl.**177**(1993), 553-571. MR**94h:32020****[ST2]**-,*Invariant Potential Theory in the Unit Ball of*, 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 -subharmonic functions*, Trans. Amer. Math. Soc.**292**(1985), 501-518. MR**87a:31007****[ZH]**S. Zhao,*On the weighted -integrability of nonnegative -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