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.

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We study the geometry of m-regular domains within the CaffarelliNirenberg-Spruck model in terms of barrier functions, envelopes, exhaustion functions, and Jensen measures. We prove among other things that every mhyperconvex domain admits an exhaustion function that is negative, smooth, strictly m-subharmonic, and has bounded m-Hessian measure.

Abstract. The aim of this paper is to give a new proof of the complete characterization of measures for which there exist a solution of the Dirichlet problem for the complex Monge-Ampère operator in the set of plurisubharmonic functions with finite pluricomplex energy. The proof uses variational methods. IntroductionThroughout this note let Ω ⊆ C n , n ≥ 1, be a bounded, connected, open, and hyperconvex set. By E 0 we denote the family of all bounded plurisubharmonic functions ϕ defined on Ω such that lim z→ξ ϕ(z) = 0 for every ξ ∈ ∂Ω , andwhere (dd c · ) n is the complex Monge-Ampère operator. Next let E p , p > 0, denote the family of plurisubharmonic functions u defined on Ω such that there exists a decreasing sequence {u j }, u j ∈ E 0 , that converges pointwise to u on Ω, as j tends to +∞, and 10,14]). It should be noted that it follows from [10] that the complex Monge-Ampère operator is well-defined on E p . It is not only within pluripotential theory these cones have been proven useful, but also as a tool in dynamical systems and algebraic geometry (see e.g. [2,17]). For further information on pluripotential theory we refer to [16,19,20]. The purpose of this paper is to give a new proof of Theorem B below and use Theorem B to prove (2) implies (1) the following theorem:Theorem A (Dirichlet's problem). Let µ be a non-negative Radon measure, then the following conditions are equivalent:(1) there exists a function u ∈ E 1 such that (dd c u) n = µ, (2) there exists a constant B > 0, such that Ω (−ϕ) dµ ≤ B e 1 (ϕ) 1 n+1 for all ϕ ∈ E 1 , (1.1) (3) the class E 1 is contained in L 1 (µ),2000 Mathematics Subject Classification. Primary 35J20; Secondary 32W20.

The Cegrell classes with zero boundary data are defined by certain decreasing approximating sequences of functions with different properties depending on the class in question. It is different for Cegrell classes which are given by a continuous function f , these classes are defined by an inequality. It is proved in this article that it is possible to define the Cegrell classes which are given by f in a similar manner as those classes with zero boundary data. An existence result for the Dirichlet problem for certain singular measures is proved. The article ends with three applications. Results connected to convergence in capacity, subextension of plurisubharmonic functions and integrability are proved.

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ModulabilityOrdered vector space Quasi-Banach space Topological dual Let p > 0, and let E p denote the cone of negative plurisubharmonic functions with finite pluricomplex p-energy. We prove that the vector space δE p = E p − E p , with the vector ordering induced by the cone E p is σ -Dedekind complete, and equipped with a suitable quasi-norm it is a non-separable quasi-Banach space with a decomposition property with control of the quasi-norm. Furthermore, we explicitly characterize its topological dual. The cone E p in the quasi-normed space δE p is closed, generating, and has empty interior.

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