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 $Omega$.
展开▼
