Separate harmonic functions, a Terada-type theorem
Let \(D\subseteq \mathbb{R}^{m}\) be a domain; \(G\subseteq \mathbb{R}^{n}\) an open set and \(E\subseteq D\) verifying a technical condition analogue of ``local Polya condition of Leja''. It is proved that, if \(f:D\times G \rightarrow C\) is such that: \[ \forall x \in E,\;f(x, \cdot) \text{ is harmonic on } G \] \[ \forall y \in G,\;f(\cdot , y) \text{ is harmonic on } D \] then \(f\) is harmonic on \(D\times G\). The case \(E = D\) was proved by \textit{P. Lelong} [Ann. Inst. Fourier 11, 515-562 (1961; Zbl 0100.07902)].
scientific article; zbMATH DE number 1436914