We give a characterization for boundedness of plurisubharmonic functions in the Cegrell class ℱ.

We consider a class of maximal plurisubharmonic functions and prove several properties of it. We also give a condition of maximality for unbounded plurisubharmonic functions in terms of the Monge-Ampère operator $\left(d{d}^c{e}^u\right)ⁿ$.

We prove a decomposition theorem for complex Monge-Ampère measures of plurisubharmonic functions in connection with their pluripolar sets.

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

The aim of this paper is to present an extension theorem for (N,k)-crosses with pluripolar singularities.

Let D be a domain in ℂⁿ. We introduce a class of pluripolar sets in D which is essentially contained in the class of complete pluripolar sets. An application of this new class to the problem of approximation of holomorphic functions is also given.

Let $\Omega \subset {ℂ}^n$ be a bounded, simply connected $ℂ$-convex domain. Let $\alpha \in {\mathbb{Z}}_{+}^n$ and let $f$ be a function on $\Omega$ which is separately $C^{2{\alpha }_j-1}$-smooth with respect to $z_j$ (by which we mean jointly $C^{2{\alpha }_j-1}$-smooth with respect to $\mathrm{Re}{z}_j$, $\mathrm{Im}{z}_j$). If $f$ is $\alpha$-analytic on $\Omega \setminus f^{-1}\left(0\right)$, then $f$ is $\alpha$-analytic on $\Omega$. The result is well-known for the case ${\alpha }_i=1$, $1\le i\le n$, even when $f$ a priori is only known to be continuous.

In this paper we obtain results on approximation, in the multidimensional complex case, of functions from ${}^{\infty }\left(K\right)$ by complex polynomials. In particular, we generalize the results of Pawłucki and Pleśniak (1986) for the real case and of Siciak (1993) in the case of one complex variable. Furthermore, we extend the results of Baouendi and Goulaouic (1971) who obtained the order of approximation in the case of Gevrey classes over real compacts with smooth analytic boundary and we present the orders of approximation...

We study boundary values of functions in Cegrell’s class ${}_{\psi }$.

Let F be the Cartesian product of N closed sets in ℂ. We prove that there exists a function g which is continuous on F and holomorphic on the interior of F such that ${\Gamma }_g\left(F\right):=(z,g(z\left)\right):z\in F$ is complete pluripolar in ${ℂ}^{N+1}$. Using this result, we show that if D is an analytic polyhedron then there exists a bounded holomorphic function g such that ${\Gamma }_g\left(D\right)$ is complete pluripolar in ${ℂ}^{N+1}$. These results are high-dimensional analogs of the previous ones due to Edlund [Complete pluripolar curves and graphs, Ann. Polon. Math. 84 (2004), 75-86]...

The energy class ${}_p$ is studied for 0 < p < 1. A characterization of certain bounded plurisubharmonic functions in terms of ${ℱ}_p$ and its pluricomplex p-energy is proved.

Let D be a domain in ℂⁿ. The plurisubharmonic envelope of a function φ ∈ C(D̅) is the supremum of all plurisubharmonic functions which are not greater than φ on D. A bounded domain D is called c-regular if the envelope of every function φ ∈ C(D̅) is continuous on D and extends continuously to D̅. The purpose of this paper is to give a complete characterization of c-regular domains in terms of Jensen measures.

Let V be an analytic variety in a domain Ω ⊂ ℂⁿ and let K ⊂ ⊂ V be a closed subset. By studying Jensen measures for certain classes of plurisubharmonic functions on V, we prove that the relative extremal function ${\omega }_K$ is continuous on V if Ω is hyperconvex and K is regular.

Let $D_j$ be a bounded hyperconvex domain in ${ℂ}^{n_j}$ and set $D=D₁\times \cdots \times D_s$, j=1,...,s, s≥ 3. Also let ₙ be the symmetrized polydisc in ℂⁿ, n ≥ 3. We characterize those real-valued continuous functions defined on the boundary of D or ₙ which can be extended to the inside to a pluriharmonic function. As an application a complete characterization of the compliant functions is obtained.

We study the relationship between convergence in capacities of plurisubharmonic functions and the convergence of the corresponding complex Monge-Ampère measures. We find one type of convergence of complex Monge-Ampère measures which is essentially equivalent to convergence in the capacity $C_n$ of functions. We also prove that weak convergence of complex Monge-Ampère measures is equivalent to convergence in the capacity $C_{n-1}$ of functions in some case. As applications we give certain stability theorems...

We prove a disc formula for the weighted Siciak-Zahariuta extremal function $V_{X,q}$ for an upper semicontinuous function q on an open connected subset X in ℂⁿ. This function is also known as the weighted Green function with logarithmic pole at infinity and weighted global extremal function.

We study the equidistribution of Fekete points in a compact complex manifold. These are extremal point configurations defined through sections of powers of a positive line bundle. Their equidistribution is a known result. The novelty of our approach is that we relate them to the problem of sampling and interpolation on line bundles, which allows us to estimate the equidistribution of the Fekete points quantitatively. In particular we estimate the Kantorovich–Wasserstein distance of the Fekete points...

Let $K$ be a compact set in an open set $D$ on a Stein manifold $\Omega$ of dimension $n$. We denote by $H^{\infty }\left(D\right)$ the Banach space of all bounded and analytic in $D$ functions endowed with the uniform norm and by $A_K^D$ a compact subset of the space $C\left(K\right)$ consisted of all restrictions of functions from the unit ball ${𝔹}_{H^{\infty }\left(D\right)}$. In 1950ies Kolmogorov posed a problem: does$${\mathscr{H}}_{\epsilon }\left(A_K^D\right)\sim \tau {\left(ln\frac{1}{\epsilon }\right)}^{n+1},\epsilon \to 0,$$where ${\mathscr{H}}_{\epsilon }\left(A_K^D\right)$ is the $\epsilon$-entropy of the compact $A_K^D$. We give here a survey of results concerned with this problem and a related problem on the strict asymptotics of Kolmogorov diameters...

We study the pluripolar hulls of analytic sets. In particular, we show that hulls of graphs of analytic functions can be multiple sheeted and sheets can be separated by a set of zero dimension.

Let $(X,\omega )$ be a compact Kähler manifold. We obtain uniform Hölder regularity for solutions to the complex Monge-Ampère equation on $X$ with $L^p$ right hand side, $p>1$. The same regularity is furthermore proved on the ample locus in any big cohomology class. We also study the range $(X,\omega )$ of the complex Monge-Ampère operator acting on $\omega$-plurisubharmonic Hölder continuous functions. We show that this set is convex, by sharpening Kołodziej’s result that measures with $L^p$-density belong to $(X,\omega )$ and proving that $(X,\omega )$ has the...