Conditions for separately subharmonic functions to be subharmonic

Abstract

Letu be a function on ℝ^m×ℝⁿ, wherem⩾2 andn⩾2, such thatu(x, .) is subharmonic on ℝⁿ for each fixedx in ℝ^m andu(.,y) is subharmonic on ℝ^m for each fixedy in ℝⁿ. We give a local integrability condition which ensures the subharmonicity ofu on ℝ^m×ℝⁿ, and we show that this condition is close to being sharp. In particular, the local integrability of (log⁺ u ⁺)^m+n−2+α is enough to secure the subharmonicity ofu if α>0, but not if α<0.