Monge–Ampère measures on pluripolar sets

Abstract

In this article we solve the complex Monge-Ampère equation for measures with large singular part. This result generalizes classical results by Demailly, Lelong and Lempert a.o., who considered singular parts carried on discrete sets. By using our result we obtain a generalization of Ko lodziej's subsolution theorem. More precisely, we prove that if a non-negative Borel measure is dominated by a complex Monge-Ampère measure, then it is a complex Monge-Ampère measure.2000 Mathematics Subject Classification. Primary 32W20; Secondary 32U15.

Subextension of plurisubharmonic functions without increasing the total Monge–Ampère mass

“…Assume that u ∈ F(H) satisfies (2.1). It follows from Proposition 2.5 in [1] that there exists a decreasing sequence [u j ] ⊂ E 0 (H) that converges pointwise to u on Ω as j → ∞. By Proposition 2.2(b) and assumption (2.1) we have…”

“…The measure µ j is a Borel measure in Ω 2 and it vanishes on pluripolar sets by Lemma 4.11 in [1]. Moreover, from (2.4) it follows that µ j (Ω 2 ) < ∞.…”

“…Hence, we can without loss of generality assume that (dd c H) n = 0. The measure (dd c u j ) n has finite total mass and vanishes on pluripolar sets by Lemma 4.11 in [1]. Therefore Lemma 5.14 of [11] implies that there exists a unique function…”

“…(a) follows directly from Lemma 3.3 in [1], and (b) from Corollary 3.4 in [1]. To prove (c) note that since if u ∈ N (H) there exists ϕ ∈ N such that H ≥ u ≥ ϕ + H and therefore H ≥ u ≥ ϕ + H, so u = H from Theorem 2.1 in [12].…”

Hölder continuous solutions to the complex Monge–Ampère equations in non-smooth pseudoconvex domains

“…Some elements of pluripotential theory that will be used throughout the paper can be found in [1]- [32]. A bounded domain Ω ⊂ C n is called hyperconvex if there exists a bounded plurisubharmonic function ρ such that {z ∈ Ω : ρ(z) < c} ⋐ Ω, for every c ∈ (−∞, 0).…”