Article

Mathematical Notes

, Volume 83, Issue 3, pp 500-505

First online:

On families of complex lines sufficient for holomorphic extension

  • A. M. KytmanovAffiliated withKrasnoyarsk State University Email author 
  • , S. G. MyslivetsAffiliated withKrasnoyarsk State University

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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 ⊂ 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.

Key words

holomorphic extension property family of complex lines Hartogs’ theorem Bochner-Martinelli integral Sard’s theorem Cauchy-Riemann condition