On families of complex lines sufficient for holomorphic extension

Abstract

It is shown that the set \( \mathfrak{L}_\Gamma \) of all complex lines passing through a germ of a generating manifold Γ is sufficient for any continuous function f defined on the boundary of a bounded domain D ⊂ ℂn with connected smooth boundary and having the holomorphic one-dimensional extension property along all lines from \( \mathfrak{L}_\Gamma \) to admit a holomorphic extension to D as a function of many complex variables.