Boundary correspondence of Nevanlinna counting functions for selfmaps of the unit disc
Authors:
Pekka J. Nieminen and Eero Saksman
Journal:
Trans. Amer. Math. Soc. 356 (2004), 31673187
MSC (2000):
Primary 30D35, 30D50; Secondary 47B33
Published electronically:
October 29, 2003
MathSciNet review:
2052945
Fulltext PDF Free Access
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Additional Information
Abstract: Let be a holomorphic selfmap of the unit disc . For every , there is a measure on (sometimes called Aleksandrov measure) defined by the Poisson representation . Its singular part measures in a natural way the ``affinity'' of for the boundary value . The affinity for values inside is provided by the Nevanlinna counting function of . We introduce a natural measurevalued refinement of and establish that the measures are obtained as boundary values of the refined Nevanlinna counting function . More precisely, we prove that is the weak limit of whenever converges to nontangentially outside a small exceptional set . We obtain a sharp estimate for the size of in the sense of capacity.
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 C. Carathéodory, Theory of Functions, Vol. II, Chelsea, New York, 1960.
 [CM]
 J. A. Cima and A. L. Matheson, Essential norms of composition operators and Aleksandrov measures, Pacific J. Math. 179 (1997), 5963. MR 98e:47047
 [F]
 S. D. Fisher, Function Theory on Planar Domains, J. Wiley & Sons, New York, 1983. MR 85d:30001
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 O. Frostman, Sur les produits de Blaschke, Kungl. Fysiog. Sällsk. i Lund Förh. 12 (1942), 169182. MR 6:262e
 [G]
 J. B. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981. MR 83g:30037
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 O. Lehto, A majorant principle in the theory of functions, Math. Scand. 1 (1953), 517. MR 15:115d
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 J. E. Littlewood, On inequalities in the theory of functions, Proc. London Math. Soc. 23 (1925), 481519.
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 J. E. Littlewood, Lectures on the theory of functions, Oxford Univ. Press, Oxford, 1944. MR 6:261f
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 R. Nevanlinna, Eindeutige analytische Funktionen, J. W. Edwards, Ann Arbor, Michigan, 1944. Second edition by SpringerVerlag, Berlin, 1953.
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 T. Ransford, Potential Theory in the Complex Plane, Cambridge Univ. Press, Cambridge, 1995. MR 96e:31001
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 W. Rudin, A generalization of a theorem of Frostman, Math. Scand. 21 (1967), 136173. MR 38:3463
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 W. Rudin, Real and Complex Analysis (3rd ed.), McGrawHill, New York, 1987.
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 D. Sarason, Composition operators as integral operators, Analysis and Partial Differential Equations, Marcel Dekker, New York, 1990. MR 92a:47040
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 D. Sarason, SubHardy Hilbert Spaces in the Unit Disk, Wiley, New York, 1995. MR 96k:46039
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 J. E. Shapiro, Aleksandrov measures used in essential norm inequalities for composition operators, J. Operator Theory 40 (1998), 133146. MR 99i:47062
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 J. H. Shapiro, The essential norm of a composition operator, Ann. Math. 125 (1987), 375404. MR 88c:47058
 [S2]
 J. H. Shapiro, Recognizing an inner function by its distribution of values, unpublished manuscript, 1999. Available at http://www.math.msu.edu/~shapiro/Pubvit/Downloads/InnerNev/InnerNev.html.
 [SS]
 J. H. Shapiro and C. Sundberg, Compact composition operators on , Proc. Amer. Math. Soc. 108 (1990), 443449. MR 90d:47035
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 C. S. Stanton, Counting functions and majorization for Jensen measures, Pacific J. Math. 125 (1986), 459468. MR 88c:32002
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Additional Information
Pekka J. Nieminen
Affiliation:
Department of Mathematics, University of Helsinki, P.O. Box 4 (Yliopistonkatu 5), FIN00014 University of Helsinki, Finland
Email:
pekka.j.nieminen@helsinki.fi
Eero Saksman
Affiliation:
Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), FIN40014 University of Jyväskylä, Finland
Email:
saksman@maths.jyu.fi
DOI:
http://dx.doi.org/10.1090/S0002994703034871
PII:
S 00029947(03)034871
Keywords:
Nevanlinna counting function,
Aleksandrov measure,
multiplicity,
boundary value,
angular derivative
Received by editor(s):
February 3, 2003
Published electronically:
October 29, 2003
Additional Notes:
The first author was supported by the Academy of Finland, project 49077
Article copyright:
© Copyright 2003
American Mathematical Society
