Plurisubharmonic approximation and boundary values of plurisubharmonic functions

Abstract

We study the problem of approximating plurisubharmonic functions on a bounded domain Ω by continuous plurisubharmonic functions defined on neighborhoods of Ω. It turns out that this problem can be linked to the problem of solving a Dirichlet type problem for functions plurisubharmonic on the compact set Ω in the sense of Poletsky. A stronger notion of hyperconvexity is introduced to fully utilize this connection, and we show that for this class of domains the duality between the two problems is perfect. In this setting, we give a characterization of plurisubharmonic boundary values, and prove some theorems regarding the approximation of plurisubharmonic functions.

Extension and approximation ofm-subharmonic functions

“…This since F ∈ SH m ( Ω), and by Theorem 2.6, these functions can be characterize by approximation on neighborhoods of Ω. If m = n, then Theorem 4.2 was proved in [30].…”

“…In the same way as in [30] we can show that it is enough to look at the measures in J m z (Ω) for z ∈ ∂Ω. Part (2) : By Theorem 2.6 part (b) we want to prove that there is a decreasing sequence of functions ϕ j in SH o m (Ω) such that ϕ j → ϕ onΩ.…”

“…The converse is not true. The case m = n was studied in [30], and discussed in [41]. A P n -hyperconvex domain is P m -hyperconvex for every m = 1, .…”

“…. , n, and as observed in [30], the notion of P n -hyperconvexity is strictly weaker than the notion of strict hyperconvexity that has been studied and used by for example Bremermann [12], and Poletsky [44]. Furthermore, a P m -hyperconvex domain is fat in the sense Ω = (Ω) • .…”

The Geometry of m-Hyperconvex Domains

“…This is possible since contains the constant functions and separates points in . Our inspiration can be traced back to the works mentioned in the introduction, but maybe more to [17] and [35].…”