We study two known theorems regarding Hermitian matrices: Bellman's principle and Hadamard's theorem. Then we apply them to problems for the complex Monge-Ampère operator. We use Bellman's principle and the theory for plurisubharmonic functions of finite energy to prove a version of subadditivity for the complex Monge-Ampère operator. Then we show how Hadamard's theorem can be extended to polyradial plurisubharmonic functions.

In our earlier paper [CKZ], we proved that any plurisubharmonic function on a bounded hyperconvex domain in ${ℂ}^n$ with zero boundary values in a quite general sense, admits a plurisubharmonic subextension to a larger hyperconvex domain. Here we study important properties of its maximal subextension and give informations on its Monge-Ampère measure. More generally, given a quasi-plurisubharmonic function $\varphi$ on a given quasi-hyperconvex domain $D\subset X$ of a compact Kähler manifold $(X,\omega )$, with well defined Monge-Ampère...

We study swept-out Monge-Ampère measures of plurisubharmonic functions and boundary values related to those measures.

Existence and uniqueness theorems for weak solutions of a complex Monge-Ampère equation are established, extending the Bedford-Taylor pluripotential theory. As a consequence, using the Tian-Yau-Zelditch theorem, it is shown that geodesics in the space of Kähler potentials can be approximated by geodesics in the spaces of Bergman metrics. Motivation from Donaldson’s program on constant scalar curvature metrics and Yau’s strategy of approximating Kähler metrics by Bergman metrics is also discussed....

We study the masses charged by ${\left(d{d}^{c}u\right)}^n$ at isolated singularity points of plurisubharmonic functions u. This is done by means of the local indicators of plurisubharmonic functions introduced in [15]. As a consequence, bounds for the masses are obtained in terms of the directional Lelong numbers of u, and the notion of the Newton number for a holomorphic mapping is extended to arbitrary plurisubharmonic functions. We also describe the local indicator of u as the logarithmic tangent to u.

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 $-\chi \left(u\right)\left(d{d}^{c}u\right)ⁿ=d\mu$. Under some additional assumption the solution u is uniquely determined.

Our aim in this article is the study of subextension and approximation of plurisubharmonic functions in ${}_{\chi }(\Omega ,H)$, the class of functions with finite χ-energy and given boundary values. We show that, under certain conditions, one can approximate any function in ${}_{\chi }(\Omega ,H)$ by an increasing sequence of plurisubharmonic functions defined on strictly larger domains.

Let μ be a non-negative measure with finite mass given by $\phi \left(d{d}^c\psi \right)ⁿ$, where ψ is a bounded plurisubharmonic function with zero boundary values and $\phi \in L^q\left(\left(d{d}^c\psi \right)ⁿ\right)$, φ ≥ 0, 1 ≤ q ≤ ∞. The Dirichlet problem for the complex Monge-Ampère operator with the measure μ is studied.

We give a pluripotential-theoretic proof of the product property for the transfinite diameter originally shown by Bloom and Calvi. The main tool is the Rumely formula expressing the transfinite diameter in terms of the global extremal function.

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 prove that plurisubharmonic solutions to certain boundary blow-up problems for the complex Monge-Ampère operator are Lipschitz continuous. We also prove that in certain cases these solutions are unique.

We give some characterizations of the class ${}_m\left(\Omega \right)$ and use them to establish a lower estimate for the log canonical threshold of plurisubharmonic functions in this class.

We find a bounded solution of the non-homogeneous Monge-Ampère equation under very weak assumptions on its right hand side.

The aim of the paper is to investigate subextensions with boundary values of certain plurisubharmonic functions without changing the Monge-Ampère measures. From the results obtained, we deduce that if a given sequence is convergent in $C_{n-1}$-capacity then the sequence of the Monge-Ampère measures of subextensions is weakly*-convergent. As an application, we investigate the Dirichlet problem for a nonnegative measure μ in the class ℱ(Ω,g) without the assumption that μ vanishes on all pluripolar sets.

We prove that subextension of certain plurisubharmonic functions is always possible without increasing the total Monge-Ampère mass.

We establish the comparison principle in the class ${}_p\left(f\right)$. The result obtained is applied to the Dirichlet problem in ${}_p\left(f\right)$.