The complex Monge-Ampère operator is a useful tool not only within pluripotential theory, but also in algebraic geometry, dynamical systems and Kähler geometry. In this self-contained survey we present a unified theory of Cegrell's framework for the complex Monge-Ampère operator.
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We prove some existence results for the complex Monge-Ampère equation $(dd^cu)ⁿ = gdλ$ in ℂⁿ in a certain class of homogeneous functions in ℂⁿ, i.e. we show that for some nonnegative complex homogeneous functions g there exists a plurisubharmonic complex homogeneous solution u of the complex Monge-Ampère equation.
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Let Ω be a bounded hyperconvex domain in ℂn and let μ be a positive and finite measure which vanishes on all pluripolar subsets of Ω. We prove that for every continuous and strictly increasing function χ:(-∞,0) → (-∞,0) there exists a negative plurisubharmonic function u which solves the Monge-Ampère type equation $-χ(u)(dd^cu)ⁿ = dμ$. Under some additional assumption the solution u is uniquely determined.
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Let $D_{j}$ be a bounded hyperconvex domain in $ℂ^{n_{j}}$ and set $D = D₁ ×⋯× D_{s}$, j=1,...,s, s ≥ 3. Also let $Ω_π$ be the image of D under the proper holomorphic map π. We characterize those continuous functions $f:∂Ω_π → ℝ$ that can be extended to a real-valued pluriharmonic function in $Ω_π$.
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Let μ be a non-negative measure with finite mass given by $φ(dd^{c}ψ)ⁿ$, where ψ is a bounded plurisubharmonic function with zero boundary values and $φ ∈ L^{q}((dd^{c}ψ)ⁿ)$, φ ≥ 0, 1 ≤ q ≤ ∞. The Dirichlet problem for the complex Monge-Ampère operator with the measure μ is studied.
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Let $D_{j}$ be a bounded hyperconvex domain in $ℂ^{n_{j}}$ and set $D = D₁ × ⋯ × D_{s}$, j=1,...,s, s≥ 3. Also let 𝔾ₙ be the symmetrized polydisc in ℂⁿ, n ≥ 3. We characterize those real-valued continuous functions defined on the boundary of D or 𝔾ₙ which can be extended to the inside to a pluriharmonic function. As an application a complete characterization of the compliant functions is obtained.
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In this note we consider radially symmetric plurisubharmonic functions and the complex Monge-Ampère operator. We prove among other things a complete characterization of unitary invariant measures for which there exists a solution of the complex Monge-Ampère equation in the set of radially symmetric plurisubharmonic functions. Furthermore, we prove in contrast to the general case that the complex Monge-Ampère operator is continuous on the set of radially symmetric plurisubharmonic functions. Finally we characterize radially symmetric plurisubharmonic functions among the subharmonic ones using merely the laplacian.
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Poletsky has introduced a notion of plurisubharmonicity for functions defined on compact sets in ℂⁿ. We show that these functions can be completely characterized in terms of monotone convergence of plurisubharmonic functions defined on neighborhoods of the compact.
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The energy class $𝓔_{p}$ is studied for 0 < p < 1. A characterization of certain bounded plurisubharmonic functions in terms of $ℱ_{p}$ and its pluricomplex p-energy is proved.
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