On families of complex lines sufficient for holomorphic extension

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 ⊂ ℂ n 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.

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