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

Theorem (with Mohammad Shirazi, McGill University).
Let $M$ be a complex manifold endowed with a distance $d$ and a regular Borel measure $\mu$, such that non-empty open sets have positive measure. Let $U \subset M$ be an arbitrary Stein domain and $\psi\in \mathcal M(\partial U)$ an arbitrary Borel measurable function on the boundary $\partial U$, whose restriction to some closed subset $S\subset\partial U$ is continuous. Then, for an arbitrary regular $\sigma$-finite Borel measure $\nu$ on $\partial U$, there exists a holomorphic function $f$ on $U$, such that, for $\nu$-almost every $p\in\partial U$, and for every $p \in S$, $f(x) \to \psi(p)$, as $x \to p$ outside a set of $\mu$-density 0 at $p$ relative to $U$.