Approximation of plurifinely plurisubharmonic functions

Abstract

In this paper, we study the approximation of negative plurifinely plurisubharmonic function defined on a plurifinely domain by an increasing sequence of plurisubharmonic functions defined in Euclidean domains.

On the approximation of weakly plurifinely plurisubharmonic functions

“…El Kadiri and Wiegerinck [18] defined the complex Monge-Ampère operator for finite F -plurisubharmonic functions on an F -domain Ω. Recently, Hong and coauthors have been successfully pushing the theory of F -plurisubharmonic functions (see [12], [13], [14], [19]). The aim of this note is to study the conditions on u and Ω such that u can be approximated by an increasing sequence of plurisubharmonic functions defined on Euclidean neighborhoods of Ω.…”

“…When Ω is bounded F -domain. In research [19], the authors gave the kind of Ω and u that are in line with the F -set up to make the approximation possible.…”

Monge–Ampère measures of $\mathcal F$-plurisubharmonic functions

“…Later on, El Kadiri and Wiegerinck [EW14] defined the complex Monge-Ampère operator on the class of finite F-plurisubharmonic functions as an operator which associates to each finite F-plurisubharmonic function on an F-open set Ω ⊂ C n a non-negative Borel measure on Ω which does not charge pluripolar sets. Recently, Trao, Viet and Hong [TVH17] studied the notion of Cegrell classes for F-plurisubharmonic functions on F-hyperconvex plurifine domains of C n . Recall that a bounded F-domain Ω ⊂ C n is called F-hyperconvex if there exists a negative bounded plurisubharmonic function γ Ω defined on a bounded hyperconvex domain The purpose of this paper is to study the pluripolar parts of complex Monge-Ampère measures of F-plurisubharmonic functions in bounded Fhyperconvex domains.…”

“…Observe that the above result tells us that P (dd c u) n vanishes outside of Ω. To provide some more properties of P (dd c u) n , we need the following notion from [TVH17]: A bounded F-hyperconvex domain Ω is said to have the F-approximation property if there exists an increasing sequence of negative plurisubharmonic functions ρ j defined on bounded hyperconvex domains Ω j such that Ω ⊂ Ω j+1 ⊂ Ω j and ρ j ρ ∈ E 0 (Ω) a.e. on Ω as j +∞.…”

“…Next, we recall from [TVH17] the definition of the Cegrell classes for F-plurisubharmonic functions.…”