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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 Cn−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.

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... Note that techniques used in the proof of the main theorem come from our results in [15,16] recently. ...

... Proof The proof is almost the same as the ones given by Hai and Hong [15]. For convenience to readers, we sketch the proof of the proposition. ...

... Proof The proof is almost the same as the ones given by Hai and Hong [15]. For readers convenience, we sketch the proof of the proposition. ...

In this paper, we study continuous ωq-plurisubharmonic exhaustion functions. It is shown that if (M, ω) is a K¨ahler manifold and Ω ⋐ M is a relatively compact open subset such that there exists a smooth strictly plurisubharmonic function onM and for every a ∈ ∂Ω, there exists a neighborhood Ua of a in M such that Ua ∩ Ω has a continuous ωq-plurisubharmonic exhaustion function then Ω has a continuous ωq-plurisubharmonic exhaustion function. © Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2015.

... Note that techniques used in the proof of the main theorem come from our results in [15,16] recently. ...

... Proof The proof is almost the same as the ones given by Hai and Hong [15]. For convenience to readers, we sketch the proof of the proposition. ...

... This Proof The proof is almost the same as the ones given by Hai and Hong [15]. For readers convenience, we sketch the proof of the proposition. ...

In this paper, we look for solutions of the complex Monge–Ampère equation with boundary values [Inline formula] in an unbounded hyperconvex domain [Inline formula] of [Inline formula].

... On the other hand, Czyz and Hed [8] studied in 2008 the subextension problem concerning to boundary values in bounded hyperconvex domains. Recently, Hai and Hong [13] proved that plurisubharmonic functions with uniformly bounded Monge-Ampère mass on a bounded hyperconvex domain always admit a plurisubharmonic subextension without changing the Monge-Ampère measures. ...

The aim of the paper is to investigate the Monge–Ampère measures of maximal subextensions of plurisubharmonic functions with given boundary values. As an application, we study the approximation of negative plurisubharmonic function with given boundary values by an increasing sequence of plurisubharmonic functions defined in larger domains.

In this paper we establish a result on subextension of m-subharmonic functions in the class F m (Ω, f) without changing the hessian measures. As application, we approximate a m-subharmonic function with given boudary value by an increasing sequence of m-subharmonic functions defined on larger domains.

In this note, we give some results on maximal subextensions of plurisubharmonic functions on hyperconvex domains in \(\mathbb C^n\) and introduce the notion about cone of maximal subextensions of plurisubharmonic functions. Furthermore, we establish the invariant of this cone through proper holomorphic surjections.

In this paper we are interested in studying the Perron–Bremermann envelope of plurisubharmonic functions. We give a sufficient condition for the envelope to be Hölder continuous.

The aim of this note is to establish a result on subextension of m-subharmonic functions in the class \(\mathcal {F}_{m}({\Omega })\) with the precise description of the complex Hessian measure of the subextend function.

The aim of the paper was to investigate subextension of plurisubharmonic functions in unbounded hyperconvex domains without changing the Monge–Ampère measures. As an application, we study approximation of plurisubharmonic functions with given boundary values in unbounded hyperconvex domains in .

In this paper, we prove the existence of weak solutions of equations of complex Monge–Ampère type for arbitrary measures, in particular, measures carried by pluripolar sets. As an application of the obtained result, we show the existence of weak solutions of equations of complex Monge–Ampère type in the class (Formula presented.) if there exist locally subsolutions.

In this paper, we study the convergence in the capacity of sequence of
plurisubharmonic functions. As an application, we prove a stability of the
solutions of the complex Monge-Amp\`ere equations

In this paper, we investigate subextension of plurisubharmonic functions in the weighted pluricomplex energy class . Moreover, we show the equality of the weighted Monge-Ampère measures of subextension and the given function.

We prove that if ε(Ω) ∋ u j → u ∈ ε(Ω) in Cn-capacity then liminf j→∞(dd cu j) n ≥ 1{u>∞}(dd cu) n. This result is used to consider the convergence in capacity on bounded hyperconvex domains and compact Kähler manifolds.

In this article, we prove that if E is a complete pluripolar set in Ω, then E = { = −∞} for some ∞(Ω). Moreover, we study the subextension in Cegrell's class p .

The complex Monge-Amp\`ere operator $(dd^c)^n$ is an important tool in complex analysis. It would be interesting to find the right notion of convergence $u_j\to u$ such that $(dd^cu_j)^n\to (dd^cu)^n$ in the weak topology. In this paper, using the $C_{n-1}$-capacity, we give a sufficient condition of the weak convergence $(dd^cu_j)^n\to (dd^cu)^n$. We also show that our condition is quite sharp in some case.

In this article we will first prove a result about convergence in capacity. Using the achieved result we will obtain a general decompositon theorem for complex Monge-Ampere measues which will be used to prove a comparison principle for the complex Monge-Ampere operator.

We define and study the domain of definition for the complex Monge-Ampere operator. This domain is the most general if we require the operator to be continuous under decreasing limits. The domain is given in terms of approximation by certain "test"-plurisubharmonic functions. We prove estimates, study of decomposition theorem for positive measures and solve a Dirichlet problem.

In this article the connection between the Cegrell classes and compliant functions is studied. A suitable norm is constructed which makes the compliant functions into a Banach space. As an application a characterization of the Dirichlet problem for pluriharmonic functions is achieved. Explicit examples of non-compliant functions will be constructed and a sufficient condition for compliance will be proved.

We collect here results on the existence and stability of weak solutions of complex Monge-Ampère equation proved by applying pluripotential theory methods and obtained in past three decades. First we set the stage introducing basic concepts and theorems of pluripotential theory. Then the Dirichlet problem for the complex Monge-Ampère equation is studied. The main goal is to give possibly detailed description of the nonnegative Borel measures which on the right hand side of the equation give rise to plurisubharmonic solutions satisfying additional requirements such as continuity, boundedness or some weaker ones. In the last part the methods of pluripotential theory are implemented to prove the existence and stability of weak solutions of the complex Monge-Ampère equation on compact Kähler manifolds. This is a generalization of the Calabi-Yau theorem. © 2005 by the American Mathematical Society. All rights reserved.

The purpose of this paper is to study convergence of Monge-Ampère measures

In this paper, we study the approximation of negative plurisubharmonic functions with given boundary values. We want to approximate a plurisubharmonic function by an increasing sequence of plurisubharmonic functions defined on strictly larger domains.

We study a general Dirichlet problem for the complex Monge-Ampère operator, with maximal plurisubharmonic functions as boundary data.

We prove that subextension of certain plurisubharmonic functions is always possible without increasing the total Monge-Ampère mass. © Instytut Matematyczny PAN, 2008.

Let Ω⋐Cn be a hyperconvex domain. Denote by E0(Ω) the class of negative plurisubharmonic functions ϕ on Ω with boundary values 0 and finite Monge–Ampère mass on Ω. Then denote by F(Ω) the class of negative plurisubharmonic functions ϕ on Ω for which there exists a decreasing sequence (ϕ)j of plurisubharmonic functions in E0(Ω) converging to ϕ such that supj∫Ω(ddcϕj)n+∞.It is known that the complex Monge–Ampère operator is well defined on the class F(Ω) and that for a function ϕ∈F(Ω) the associated positive Borel measure is of bounded mass on Ω. A function from the class F(Ω) is called a plurisubharmonic function with bounded Monge–Ampère mass on Ω.We prove that if Ω and Ω are hyperconvex domains with Ω⋐Ω⋐Cn and ϕ∈F(Ω), there exists a plurisubharmonic function ϕ̃∈F(Ω) such that ϕ̃⩽ϕ on Ω and ∫Ω(ddcϕ̃)n⩽∫Ω(ddcϕ)n. Such a function is called a subextension of ϕ to Ω.From this result we deduce a global uniform integrability theorem for the classes of plurisubharmonic functions with uniformly bounded Monge–Ampère masses on Ω.To cite this article: U. Cegrell, A. Zeriahi, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

We study the problem of approximating plurisubharmonic functions on a bounded
domain $\Omega$ by continuous plurisubharmonic functions defined on
neighborhoods of $\bar\Omega$. It turns out that this problem can be linked to
the problem of solving a Dirichlet type problem for functions plurisubharmonic
on the compact set $\bar\Omega$ in the sense of Poletsky. A stronger notion of
hyperconvexity is introduced to fully utilize this connection, and we show that
for this class of domains the duality between the two problems is perfect. In
this setting, we give a characterization of plurisubharmonic boundary values,
and prove some theorems regarding the approximation of plurisubharmonic
functions.

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.

In this article we solve the complex Monge–Ampère problem 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 Kołodziej'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.RésuméDans cet article, nous résolvons le problème de Monge–Ampère complexe pour des mesures ayant une grande partie singulière. Ce résultat généralise des résultats classiques de Demailly, Lelong et Lempert entre autres, qui considéraient des parties singulières portées par des ensembles discrets. En utilisant notre résultat, nous obtenons une généralisation du théorème de la sous-solution de Kołodziej. De manière plus précise, nous montrons que si une mesure de Borel non négative est dominée par une mesure complexe de Monge–Ampère, alors c'est une mesure complexe de Monge–Ampère.

In this article we will first prove a result about the convergence in capacity. Next we will obtain a general decomposition theorem for complex Monge-Ampère measures which will be used to prove a comparison principle for the complex Monge-Ampère operator.

Vietnam E-mail: mauhai@fpt.vn xuanhongdhsp@yahoo.com Received 5.6.2013 and in final form 2

- Le Mau

Le Mau Hai, Nguyen Xuan Hong
Department of Mathematics
Hanoi National University of Education
Hanoi, Vietnam
E-mail: mauhai@fpt.vn
xuanhongdhsp@yahoo.com
Received 5.6.2013
and in final form 2.7.2013
(3128)

- P H Hiep

P. H. Hiep, Convergence in capacity, Ann. Polon. Math. 93 (2008), 91-99.