Local Fatou Theory for Meromorphic Functions and Maps

We study admissible boundary limits of meromorphic functions and maps on strongly pseudoconvex domains through the Ahlfors–Shimizu characteristic measure δf∗ωFS ∧χ. On an open boundary set E, finiteness of this measure on shallow collars is equivalent to local integrability of the admissible spherical area function and to the local Nevanlinna class, and it yields finite admissible limits almost everywhere on E. Pointwise control does not suffice: in every dimension there are holomorphic func- tions with bounded characteristic in admissible regions at every boundary point but with finite admissible limits almost nowhere. Guided by Stein’s theorem, we propose a meromorphic equivalence between admissible limits, admissible boundedness away from a value, finiteness of the spherical area function, and bounded characteristic on sawtooth regions. In one variable, combining the classical theorems of Plessner and Spencer with a Privalov-domain localization, we show that these conditions are equiv- alent to bounded characteristic on Privalov domains, for functions and for maps of the disc. In several variables we prove part of the equivalence and obtain limits along transversal complex lines.