Boundary correspondence of Nevanlinna counting functions for self-maps of the unit disc
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- by Pekka J. Nieminen and Eero Saksman PDF
- Trans. Amer. Math. Soc. 356 (2004), 3167-3187 Request permission
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.References
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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
- MR Author ID: 315983
- Email: saksman@maths.jyu.fi
- 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
- © Copyright 2003 American Mathematical Society
- 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
- MathSciNet review: 2052945