On separately subharmonic functions (Lelong's problem)
We will call a function \(u:\mathbb{R}^{n}\times \mathbb{R}^{m}\rightarrow \mathbb{R}\) separately subharmonic-harmonic if \(u(\cdot ,y)\) is subharmonic for each \(y\), and \(u(x,\cdot )\) is harmonic for each \(x\). The author shows that \(u\) can then be represented locally in the form \(u^{\ast }+U\), where \(U\) is jointly subharmonic, \(u^{\ast }\) is separately subharmonic-harmonic, and \(u^{\ast }\) is jointly harmonic outside some nowhere dense set. The question of whether \(u\) itself is jointly subharmonic remains open.
scientific article; zbMATH DE number 5956568