Quasi-Nearly Subharmonicity and Separately Quasi-Nearly Subharmonic Functions

Wiegerinck has shown that a separately subharmonic function need not be subharmonic. Improving previous results of Lelong, Avanissian, Arsove, and of us, Armitage and Gardiner gave an almost sharp integrability condition which ensures a separately subharmonic function to be subharmonic. Completing now our recent counterparts to the cited results of Lelong, Avanissian and Arsove for so-called quasi-nearly subharmonic functions, we present a counterpart to the cited result of Armitage and Gardiner for separately quasinearly subharmonic function. This counterpart enables us to slightly improve Armitage’s and Gardiner’s original result, too. The method we use is a rather straightforward and technical, but still by no means easy, modification of Armitage’s and Gardiner’s argument combined with an old argument of Domar.