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# Arkiv för Matematik

## Volume 55 (2017)

### Number 1

### A note on approximation of plurisubharmonic functions

Pages: 229 – 241

DOI: http://dx.doi.org/10.4310/ARKIV.2017.v55.n1.a12

#### Authors

#### 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.

#### Keywords

plurisubharmonic function, approximation, Mergelyan type approximation

#### 2010 Mathematics Subject Classification

Primary 32U05. Secondary 31B05, 31B25.

Received 5 October 2016

Received revised 27 January 2017

Published 26 September 2017