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Boundary correspondence of Nevanlinna counting functions for self-maps of the unit disc


Authors: Pekka J. Nieminen and Eero Saksman
Journal: Trans. Amer. Math. Soc. 356 (2004), 3167-3187
MSC (2000): Primary 30D35, 30D50; Secondary 47B33
DOI: https://doi.org/10.1090/S0002-9947-03-03487-1
Published electronically: October 29, 2003
MathSciNet review: 2052945
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Abstract: Let $\phi$ be a holomorphic self-map of the unit disc $\mathbb{D}$. For every $\alpha \in \partial\mathbb{D}$, there is a measure $\tau_\alpha$ on $\partial\mathbb{D}$ (sometimes called Aleksandrov measure) defined by the Poisson representation $\operatorname{Re}(\alpha+\phi(z))/(\alpha-\phi(z)) = \int P(z,\zeta) \,d\tau_\alpha(\zeta)$. Its singular part $\sigma_\alpha$ measures in a natural way the ``affinity'' of $\phi$ for the boundary value $\alpha$. The affinity for values $w$ inside $\mathbb{D}$ is provided by the Nevanlinna counting function $N(w)$ of $\phi$. We introduce a natural measure-valued refinement $M_w$ of $N(w)$ and establish that the measures $\{\sigma_\alpha\}_{\alpha\in\partial\mathbb{D}}$are obtained as boundary values of the refined Nevanlinna counting function $M$. More precisely, we prove that $\sigma_\alpha$ is the weak$^*$ limit of $M_w$ whenever $w$ converges to $\alpha$non-tangentially outside a small exceptional set $E$. We obtain a sharp estimate for the size of $E$ in the sense of capacity.


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Additional Information

Pekka J. Nieminen
Affiliation: Department of Mathematics, University of Helsinki, P.O. Box 4 (Yliopistonkatu 5), FIN-00014 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), FIN-40014 University of Jyväskylä, Finland
Email: saksman@maths.jyu.fi

DOI: https://doi.org/10.1090/S0002-9947-03-03487-1
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

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