On smoothing of plurisubharmonic functions on unbounded domains

Abstract

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