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.

A Fréchet space with a sequence ${{\left|\right|·\left|\right|}_k}_{k=1}^{\infty }$ of generating seminorms is called tame if there exists an increasing function σ: ℕ → ℕ such that for every continuous linear operator T from into itself, there exist N₀ and C > 0 such that ${\left|\right|T\left(x\right)\left|\right|ₙ\le C\left|\right|x\left|\right|}_{\sigma \left(n\right)}$ ∀x ∈ , n ≥ N₀. This property does not depend upon the choice of the fundamental system of seminorms for and is a property of the Fréchet space . In this paper we investigate tameness in the Fréchet spaces (M) of analytic functions on Stein manifolds M equipped with the compact-open...

We prove some existence results for the complex Monge-Ampère equation $\left(d{d}^{c}u\right)ⁿ=gd\lambda$ 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.

We define and study the domain of definition for the complex Monge-Ampère 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.

We show that if a decreasing sequence of subharmonic functions converges to a function in $W_{loc}^{1,2}$ then the convergence is in $W_{loc}^{1,2}$.

We prove that the image of a finely holomorphic map on a fine domain in ℂ is a pluripolar subset of ℂⁿ. We also discuss the relationship between pluripolar hulls and finely holomorphic functions.

We calculate the transfinite diameter for the real unit ball $B_d:=x\in {ℝ}^d:\left|x\right|\le 1$ and the real unit simplex $T_d:=x\in {ℝ}_{+}^d:{\sum }_{j=1}^d{x}_j\le 1.$

This paper is concerned with the problem of extension of separately holomorphic mappings defined on a "generalized cross" of a product of complex analytic spaces with values in a complex analytic space. The crosses considered here are inscribed in Borel rectangles (of a product of two complex analytic spaces) which are not necessarily open but are non-pluripolar and can be quite small from the topological point of view. Our first main result says that the singular...

We prove some existence results for equations of complex Monge-Ampère type in strictly pseudoconvex domains and on Kähler manifolds.