Hartogs-Osgood theorem for separately harmonic functions. (English) Zbl 1129.31002

A function \(h(x,y)\), where \(x\in \mathbb{R}^{k}\) and \(y\in \mathbb{R}^{m-k}\) , is called separately harmonic if \(h(x,\cdot )\) and \(h(\cdot ,y)\) are each harmonic where defined for any choice of \(x\) and \(y\). Now let \(D\) be a bounded domain in \(\mathbb{R}^{k}\times \mathbb{R}^{m-k}\) such that \(\partial D\) is an \((m-1)\)-dimensional submanifold of \(\mathbb{R}^{m}\). The author shows that, if \(h\) is separately harmonic on an open neighbourhood of \(\partial D\), then \(h\) can be extended to a separately harmonic function on \(D\).


31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
32U99 Pluripotential theory
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