A note on approximation of plurisubharmonic functions

Abstract

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.