Quasi-nearly subharmonicity and separately quasi-nearly subharmonic functions

TL;DR

This paper extends the theory of quasi-nearly subharmonic functions, providing conditions under which separately quasi-nearly subharmonic functions are globally quasi-nearly subharmonic, improving previous integrability results.

Contribution

It offers a new counterpart to existing results, enhancing the understanding of when separately quasi-nearly subharmonic functions are globally quasi-nearly subharmonic.

Findings

Established a new integrability condition for quasi-nearly subharmonic functions

Improved previous results by Armitage and Gardiner

Extended the theory to separately quasi-nearly subharmonic functions

Abstract

Wiegerinck has shown that a separately subharmonic function need not be subharmonic. Improving previous results of Lelong, of Avanissian, of 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 quasi-nearly subharmonic functions. 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.