Asymptotic first boundary value problem for holomorphic functions of several complex variables

TL;DR

This paper extends classical boundary value problems for holomorphic functions from the unit disc to Stein domains on complex manifolds, analyzing asymptotic boundary behavior and limits.

Contribution

It introduces an asymptotic boundary value problem for holomorphic functions on Stein domains, generalizing classical results to complex manifolds.

Findings

Established existence of boundary limits in the asymptotic sense

Generalized Lehto's theorem to Stein domains on complex manifolds

Provided new techniques for boundary behavior analysis in several complex variables

Abstract

In 1955, Lehto showed that, for every measurable function $\psi$ on the unit circle $\mathbb T,$ there is function $f$ holomorphic in the unit disc $\mathbb D,$ having $\psi$ as radial limit a.e. on $\mathbb T.$ We consider an analogous boundary value problem, where the unit disc is replaced by a Stein domain on a complex manifold and radial approach to a boundary point $p$ is replaced by (asymptotically) total approach to $p.$