A note on approximation of plurisubharmonic functions

TL;DR

This paper investigates conditions under which continuous plurisubharmonic functions can be approximated by functions that are plurisubharmonic on neighborhoods, extending previous results and providing examples of limitations.

Contribution

It extends existing approximation results for plurisubharmonic functions to domains with less regular boundaries and identifies cases where approximation fails.

Findings

Approximation is possible if the boundary is $C^0$ outside a countable set.

Approximation works for H"older continuous functions under weaker boundary conditions.

Examples show that approximation can fail even when it is possible for H"older continuous functions.

Abstract

We extend a recent result of Avelin, Hed, and Persson about approximation of functions $u$ that are plurisubharmonic on a domain $\Omega$ and continuous on $\bar\Omega$, with functions that are plurisubharmonic on (shrinking) neighborhoods of $\bar\Omega$. 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 H\"older 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 H\"older continuous case is possible.