Families of subharmonic functions and separately subharmonic functions

ABSTRACT

We prove that the upper envelope of a family of subharmonic functions defined on an open subset of ${\mathbb{R}}^N$, $(N\ge 2)$, that is finite every where, is locally bounded above outside a closed nowhere dense set with no bounded components. Then we conclude as a consequence that a separately subharmonic function is subharmonic outside a closed nowhere dense set with no bounded components. It generalizes a result due to Cegrell and Sadullaev.

COMMUNICATED BY:

• J. Leiterer

KEYWORDS:

• Family of subharmonic functions
• Separately subharmonic functions

AMS SUBJECT CLASSIFICATIONS:

• 31 XX
• 31 AXX
• 31 BXX

Acknowledgments

The author wishes to express his gratitude to Professor Ahmed Zeriahi who introduced the author to the fascinating world of potential theory, quiet some times ago. It is a pleasure to thank Professor Azimbay Sadullaev for his help and advice during the past years to present. This is also a pleasure to thank Professor Juhani Riihentaus for his generous help to the author during the last long years.

Disclosure statement

No potential conflict of interest was reported by the author.