Boundary correspondence of Nevanlinna counting functions for self-maps of the unit disc期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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

Authors:	Pekka J Nieminen Eero Saksman

Institution:	Department of Mathematics, University of Helsinki, P.O. Box 4 (Yliopistonkatu 5), FIN-00014 University of Helsinki, Finland ; Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), FIN-40014 University of Jyväskylä, Finland

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 inside $\mathbb{D}$ is provided by the Nevanlinna counting function of $\phi$ . We introduce a natural measure-valued refinement of and establish that the measures $\{\sigma_\alpha\}_{\alpha\in\partial\mathbb{D}}$ are obtained as boundary values of the refined Nevanlinna counting function . More precisely, we prove that $\sigma_\alpha$ is the weak limit of whenever converges to $\alpha$ non-tangentially outside a small exceptional set . We obtain a sharp estimate for the size of in the sense of capacity.

Keywords:	Nevanlinna counting function Aleksandrov measure multiplicity boundary value angular derivative

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