On smoothing of plurisubharmonic functions on unbounded domains

TL;DR

This paper investigates the smoothing properties of plurisubharmonic functions on unbounded pseudoconvex domains, revealing differences in core sets for various smoothness classes and constructing functions with specific approximation properties.

Contribution

It demonstrates the existence of pseudoconvex domains where the core sets differ for different smoothness levels and constructs bounded continuous plurisubharmonic functions not approximable by smoother ones.

Findings

Existence of domains with strict inclusion of core sets for different smoothness classes

Construction of bounded continuous plurisubharmonic functions not approximable by $ ext{C}^1$-smooth functions

Illustration of limitations in smoothing of plurisubharmonic functions on unbounded domains

Abstract

We prove that for every $n \ge 2$, there exists a pseudoconvex domain $\Omega \subset \mathbb{C}^n$ such that $\mathfrak{c}^0(\Omega) \subsetneq \mathfrak{c}^1(\Omega)$, where $\mathfrak{c}^k(\Omega)$ denotes the core of $\Omega$ with respect to $\mathcal{C}^k$-smooth plurisubharmonic functions on $\Omega$. Moreover, we show that there exists a bounded continuous plurisubharmonic function on $\Omega$ that is not the pointwise limit of a sequence of $\mathcal{C}^1$-smooth bounded plurisubharmonic functions on $\Omega$.