On 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 ${\mathfrak{c}}^0(\Omega )⊊{\mathfrak{c}}^1(\Omega )$, 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 $\Omega$.