The range of the complex Monge-Ampere operator II

Abstract

Abstract. If ν is a complex Monge-Ampère mass of a bounded plurisubharmonic function then so is every nonnegative Borel measure µ ≤ ν.

Weighted Pluricomplex Energy II

“…Such measures dominated by the Monge-Ampère capacity have been extensively studied by Kołodziej in [3][4][5]. He proved that if : Ω → R is a continuous function and ∫ …”

“…Therefore the question of describing the measures which are the Monge-Ampère of bounded psh functions is very important for pluripotential theory, complex dynamic, and complex geometry. This problem has been studied extensively by various authors; see, for example, [2][3][4][5][6] and reference therein. In [7], Cegrell introduced the pluricomplex energy classes E (Ω) and F (Ω) ( ≥ 1) on which the complex Monge-Ampère operator is well defined.…”

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

“…When ϕ is merely bounded subsolution, the subsolution theorem in [16] has provided a unique bounded solution. Thus, to answer Zeriahi's question it remains to show the Hölder continuity of the bounded solution.…”

“…However, there is still an open question to find a characterisation for the measures admitting Hölder continuous solutions to the equation. If one requires only bounded solutions, then Kołodziej's subsolution theorem [16] gives such a criterion. Let ϕ ∈ P S H( ) ∩ C 0,α (¯ ) with 0 < α ≤ 1, where C 0,α (¯ ) stands for the set of Hölder continuous functions of exponent α in¯ .…”

Kähler-Einstein metrics with mixed Poincaré and cone singularities along a normal crossing divisor

“…[23], [28] e.g). In the conic case, Kołodziej's theorem [24] ensures that the potentials we are dealing with are continuous, and the uniqueness is then a consequence of the classical comparison principle established in [2, Theorem 4.1].…”