A note on approximation of plurisubharmonic functions

We extend a recent result of Avelin, Hed, and Persson about approximation of functions $f$ that are plurisubharmonic on a domain $\Omega$ and continuous on $\overline{\Omega}$, with functions that are plurisubharmonic on (shrinking) neighborhoods of $\overline{\Omega}$. We show that such approximation is possible if the boundary of $\Omega$ is $C^0$ outside a countable exceptional set $E \subset \partial \Omega$. In particular, approximation is possible on the Hartogs triangle. For Hölder continuous $u$, approximation is possible under less restrictive conditions on $E$. We next give examples of domains where this kind of approximation is not possible, even when approximation in the Hölder continuous case is possible.

Keywords

plurisubharmonic function, approximation, Mergelyan type approximation

2010 Mathematics Subject Classification

Primary 32U05. Secondary 31B05, 31B25.

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Received 5 October 2016

Received revised 27 January 2017

Accepted 8 February 2017

Published 26 September 2017