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3 Annales Polonici Mathematici

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The complex Monge-Ampère operator in the Cegrell classes

100%

Czyż R.

Instytut Matematyczny Polskiej Akademii Nauk

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.

The complex Monge-Ampère equation for complex homogeneous functions in ℂⁿ

92%

Czyż R.

nr 3

287-302

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.

Pluriharmonic extension in proper image domains

92%

Czyż R.

2009

tom 96

nr 2

163-174

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 $Ω_π$.

On the Dirichlet problem in the Cegrell classes

58%

Czyż R. , Åhag P.

nr 3

273-279

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.

Kolodziej's subsolution theorem for unbounded pseudoconvex domains

58%

Åhag P. , Czyż R.

tom 50

7-23

In this paper we generalize Kolodziej's subsolution theorem to bounded and unbounded pseudoconvex domains, and in that way we are able to solve complex Monge-Ampère equations on general pseudoconvex domains. We then give a negative answer to a question of Cegrell and Kolodziej by constructing a compactly supported Radon measure $\mu$ that vanishes on all pluripolar sets in $C^n$ such that $\lambda(C^n)=(2\pi)^n$, and for which there is no function $u$ in $\mathcal L_+$ such that $(dd^cu)^n=\mu$. We end this paper by solving a Monge_Amp±re type equation. Furthermore, we prove uniqueness and stability of the solution.