The complex Monge-Ampère equation in the Cegrell's classes
Hoang Nhat Quy
本文研究Cegrell类中复Monge-Ampère方程，给出了在$\mathcal E_χ(Ω)$与$\mathcal E_{χ,loc}(Ω)$中解存在性的精确刻画。
本文给出了在类$\mathcal E_χ(Ω)$和$\mathcal E_{χ,loc}(Ω)$中复Monge-Ampère方程解的存在性的若干精确刻画。
In this paper, we give some precise characterizations of existence of solution to the complex Monge - Ampère equation in the classes $\mathcal E_χ(Ω)$ and $\mathcal E_{χ,loc}(Ω)$.
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In this paper we prove an existence and uniqueness result for a Monge-Ampère type equation in the Cegrell class $\mathcal{E}_χ$.
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In this paper we study the relation between the weighted energy class $\mathcal{E}_χ$ introduced by S. Benelkouchi, V. Guedj and A. Zeriahi recently with the classes $\mathcal{E}$ and $\mathcal{N}$ studied by Cegrell. Moreover, we establish a generalized comparison principle for the operator $\text{M}_χ$ and, as an application, we prove a slight version of existence of solutions of Monge-Ampère type equation in the class $\mathcal{E}_χ(H,Ø)$.
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Let $X$ be a compact Kähler manifold and $\om$ a smooth closed form of bidegree $(1,1)$ which is nonnegative and big. We study the classes ${\mathcal E}_χ(X,\om)$ of $\om$-plurisubharmonic functions of finite weighted Monge-Ampère energy. When the weight $χ$ has fast growth at infinity, the corresponding functions are close to be bounded. We show that if a positive Radon measure is suitably dominated by the Monge-Ampère capacity, then it belongs to the range of the Monge-Ampère operator on some class ${\mathcal E}_χ(X,\om)$. This is done by establishing a priori estimates on the capacity of su…
We study geodesics for plurisubharmonic functions from the Cegrell class ${\mathcal F}_1$ on a bounded hyperconvex domain of ${\mathbb C}^n$ and show that, as in the case of metrics on Kähler compact menifolds, they linearize an energy functional. As a consequence, we get a uniqueness theorem for functions from ${\mathcal F}_1$ in terms of total masses of certain mixed Monge-Ampère currents. Geodesics of relative extremal functions are considered and a reverse Brunn-Minkowski inequality is proved for capacities of multiplicative combinations of multi-circled compact sets. We also show that fun…
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本文研究了退化四元数Monge-Ampère方程在加权能量类中的性质，证明了算子的良定义性、Cegrell型类的精细性质及质量集中定理，并刻画了算子的有限能量范围。
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The main goal of this article is to find, following the approach given in [Ce1] and [Ce2], the largest possible sub-class of plurisubharmornic functions on a complex variety on which the complex Monge-Ampère operator can be reasonably defined. Moreover, we are also interested in giving sufficient condition for solvability of the Monge-Ampère operator Monge-Ampère operators on complex varieties.
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Higher Lelong numbers versus full Monge-Ampère mass
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