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 compare the yields of two methods to obtain Bernstein type pointwise estimates for the derivative of a multivariate polynomial in a domain where the polynomial is assumed to have sup norm at most 1. One method, due to Sarantopoulos, relies on inscribing ellipses in a convex domain K. The other, pluripotential-theoretic approach, mainly due to Baran, works for even more general sets, and uses the pluricomplex Green function (the Zaharjuta-Siciak extremal function). When the inscribed ellipse method...

We compute the constant sup $(1/degP)\left(ma{x}_{S}log\right|P|-{\int }_{S}log|P\left|d\sigma \right)$ : P a polynomial in ${ℂ}^n$, where S denotes the euclidean unit sphere in ${ℂ}^n$ and σ its unitary surface measure.

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

We give an elementary proof of the product formula for the multivariate transfinite diameter using multivariate Leja sequences and an identity on vandermondians.

Let $G=K^{ℂ}$ be a complex reductive group. We give a description both of domains $\Omega \subset G$ and plurisubharmonic functions, which are invariant by the compact group, $K$, acting on $G$ by (right) translation. This is done in terms of curvature of the associated Riemannian symmetric space $M:=G/K$. Such an invariant domain $\Omega$ with a smooth boundary is Stein if and only if the corresponding domain ${\Omega }_M\subset M$ is geodesically convex and the sectional curvature of its boundary $S:=\partial {\Omega }_M$ fulfills the condition $K^S\left(E\right)\ge K^M\left(E\right)+k(E,n)$. The term $k(E,n)$ is explicitly computable...

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

Let K be any subset of ${ℂ}^N$. We define a pluricomplex Green’s function $V_{K,\theta }$ for θ-incomplete polynomials. We establish properties of $V_{K,\theta }$ analogous to those of the weighted pluricomplex Green’s function. When K is a regular compact subset of ${ℝ}^N$, we show that every continuous function that can be approximated uniformly on K by θ-incomplete polynomials, must vanish on $K\setminus supp{\left(d{d}^c{V}_{K,\theta }\right)}^N$. We prove a version of Siciak’s theorem and a comparison theorem for θ-incomplete polynomials. We compute $supp{\left(d{d}^c{V}_{K,\theta }\right)}^N$ when K is a compact section.

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

For a regular, compact, polynomially convex circled set $K$ in ${\mathbf{C}}^2$, we construct a sequence of pairs $\{P_n,Q_n\}$ of homogeneous polynomials in two variables with $\mathrm{deg}{P}_n=$$\mathrm{deg}{Q}_n...$

The goal of this paper is to extend the concepts of algebraic and Liouville currents, previously defined for positive closed currents by M. Blel, S. Mimouni and G. Raby, to psh currents on ${ℂ}^n$. Thus, we study the growth of the projective mass of positive currents on ${ℂ}^n$ whose support is contained in a tubular neighborhood of an algebraic subvariety. We also give a sufficient condition guaranteeing that a negative psh current is Liouville. Moreover, we prove that every negative psh algebraic current...

We prove that an analytic surface $V$ in a neighborhood of the origin in ${ℂ}^3$ satisfies the local Phragmén-Lindelöf condition ${\mathrm{PL}}_{\mathrm{loc}}$ at the origin if and only if $V$ satisfies the following two conditions: (1) $V$ is nearly hyperbolic; (2) for each real simple curve $\gamma$ in ${ℝ}^3$ and each $d\ge 1$, the (algebraic) limit variety $T_{\gamma ,d}V$ satisfies the strong Phragmén-Lindelöf condition. These conditions are also necessary for any pure $k$-dimensional analytic variety $V$ to satisify ${\mathrm{PL}}_{\mathrm{loc}}$.

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.

We give a simplified proof of J. P. Rosay's result on plurisubharmonicity of the envelope of the Poisson functional [10].