Pluricomplex energy

Abstract

Proof. Since uEC1, it follows from Lemma 3.4 that vjEC1, VjEN, and there is a decreasing sequence uj EC0 with lim uj --u Then f and sup ] -uj (ddCuj) n = ~ < +oc. J J max(uj, Vk)(ddCmax(uj, Vk)) n >//uj(ddCuj) n >1 -a, Vj, k E N, so it is enough to prove that lim Juj(ddCuj)n=/u(ddCu) n. j~+cr We have for k>~j, f -uj(ddCuj)n <~ J-uj(dd~uk) ~ = f~ -uj(dd~uk)n+/ -uj(dd~uk) n j >~--e J uj <-e SO sup j(d+( u~+U(O,-f)))~ = ~ < +~.

On the Hölder continuous subsolution problem for the complex Monge–Ampère equation

“…The following basic properties of E 0 and E 0 which will be used later, and we refer the readers to [6,7] for detailed proofs.…”

“…In what follows we always denote for p > 0 6) where dV 2n is the Lebesgue measure in C n . The constant C > 0 will appear in many places below, we understand that it is a uniform constant and it may differ from place to place.…”

Boundary values of functions in Cegrell's class Eψ

“…The following classes of psh functions were introduced by Cegrell in [Ce1] and [Ce2]: Cegrell [Ce3] introduced a new class of psh functions…”

“…He discovered the following interesting formula: [Åh], [Ce1,2], [ÅCH], [H1,2]). By Theorem 5.8 in [Ce2] we find a function u ∈ F 1 such that lim inf z→ξ u(z) = −∞ for all ξ ∈ ∂Ω.…”

A characterization of bounded plurisubharmonic functions

“…The complex Monge-Ampère operator (dd c ) n is well defined over the class of locally bounded psh functions, according to the fundamental work of BedfordTaylor in [BT1], [BT2]. Recently Cegrell has introduced in [Ce1], [Ce2] new classes of psh functions on which the complex Monge-Ampère operator can be defined and enjoys important properties, e.g. is continuous under monotone sequences.…”