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

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.