6533b7d6fe1ef96bd1265a11

RESEARCH PRODUCT

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

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Let ϕ \phi be a holomorphic self-map of the unit disc D \mathbb {D} . For every α ∈ ∂ D \alpha \in \partial \mathbb {D} , there is a measure τ α \tau _\alpha on ∂ D \partial \mathbb {D} (sometimes called Aleksandrov measure) defined by the Poisson representation Re ⁡ ( α + ϕ ( z ) ) / ( α − ϕ ( z ) ) = ∫ P ( z , ζ ) d τ α ( ζ ) \operatorname {Re}(\alpha +\phi (z))/(\alpha -\phi (z)) = \int P(z,\zeta ) \,d\tau _\alpha (\zeta ) . Its singular part σ α \sigma _\alpha measures in a natural way the “affinity” of ϕ \phi for the boundary value α \alpha . The affinity for values w w inside D \mathbb {D} is provided by the Nevanlinna counting function N ( w ) N(w) of ϕ \phi . We introduce a natural measure-valued refinement M w M_w of N ( w ) N(w) and establish that the measures { σ α } α ∈ ∂ D \{\sigma _\alpha \}_{\alpha \in \partial \mathbb {D}} are obtained as boundary values of the refined Nevanlinna counting function M M . More precisely, we prove that σ α \sigma _\alpha is the weak ∗ ^* limit of M w M_w whenever w w converges to α \alpha non-tangentially outside a small exceptional set E E . We obtain a sharp estimate for the size of E E in the sense of capacity.