Some Remarks on Approximation of Plurisubharmonic Functions

Let Ω be a domain in Cn. An upper semicontinuous function u : Ω → [−∞,∞) is said to be plurisubharmonic if the restriction of u to each complex line is subharmonic (we allow the function identically −∞ to be plurisubharmonic). We say that u is strictly plurisubharmonic if for every z0 ∈ Ω there is a neigbourhood U of z0 and λ > 0 such that u(z) − λ|z|2 is plurisubharmonic on U . We write PSH(Ω) for the set of plurisubharmonic functions on Ω,PSH−(Ω) for the subset of bounded from above and PSHc(Ω) for the set of continuous functions on Ω which are plurisubharmonic on Ω. It is a natural question whether it is possible to approximate elements of one of these classes by elements of a smaller one. It may be useful to recall some known facts regarding this problem. Richberg [5] showed that for every strictly plurisubharmonic continuous function u on Ω and every positive continuous function ε on Ω, there exists a C∞ smooth strictly plurisubharmonic ũ on Ω such that 0 < u− ũ < ε on Ω. Later, Fornaess and Narasimhan proved that every plurisubharmonic function u on a pseudoconvex domain Ω can be approximated from above by a sequence of C∞ smooth strictly plurisubharmonic functions. A remarkable example of Fornaess [1] shows that the above conclusion fails without the assumption on pseudoconvexity of Ω. In fact, the domain Ω in Fornaess’example is a smoothly bounded Hartogs domain in C2. Recall that Ω is said to be Hartogs if (z,w) ∈ Ω ⇒ (z,w′) ∈ Ω provided that |w| = |w′|. Regarding approximation on smoothly bounded domain, Fornaess and Wiegerinck [2] proved that every continuous function on Ω which is plurisubharmonic on Ω, can be approximated uniformly on Ω by C∞ smooth plurisubharmonic functions on neigbourhoods of Ω. Besides, Fornaess and Wiegerinck show that every plurisubharmonic function u on a bounded Reinhardt domain Ω can be approximated from above by a sequence of smooth strictly plurisubharmonic functions on Ω. Recall that Ω is said to be Reinhardt if (z1, · · · , zn) ∈ Ω ⇒ (z′ 1, · · · , z′ n) ∈ Ω provided that |z′ i| = |zi| for every 1 i n. This result is very interesting in comparison with the mentioned above example of Fornaess. The aim of the present paper is to study the problem of approximation on Ω of the upper regularization u∗ of a given function u ∈ PSH−(Ω) by elements in PSHc(Ω), where u∗(z) = limsup ξ→z u(ξ). This problem has been considered by in [7]