Let $h$ be a separately harmonic function on an open neighborhood of a $(m-1)$-dimensional compact submanifold $\Sigma$ in R$^m$ with $m\geq 2$. We show that $h$ can be extended to a separately harmonic function on the bounded component of R$^m-\Sigma$.
"Hartogs-Osgood theorem for separately harmonic functions." Proc. Japan Acad. Ser. A Math. Sci. 83 (2) 16 - 18, March 2007. https://doi.org/10.3792/pjaa.83.16