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.

The aim of this paper is to establish the equivalence between the non-pluripolarity of a compact set in a complex space and the property $\left(L{B}^{\infty }\right)$ for the dual space of the space of germs of holomorphic functions on that compact set.

Let D be a smooth bounded pseudoconvex domain in ℂⁿ of finite type. We prove an estimate on the pluricomplex Green function ${}_D(z,w)$ of D that gives quantitative information on how fast the Green function vanishes if the pole w approaches the boundary. Also the Hölder continuity of the Green function is discussed.

Let $X$ be a submanifold of a manifold $Z$. We address the question: When do viscosity subsolutions of a fully nonlinear PDE on $Z$, restrict to be viscosity subsolutions of the restricted subequation on $X$? This is not always true, and conditions are required. We first prove a basic result which, in theory, can be applied to any subequation. Then two definitive results are obtained. The first applies to any “geometrically defined” subequation, and the second to any subequation which can be transformed...

We establish disc formulas for the Siciak-Zahariuta extremal function of an arbitrary open subset of complex affine space. This function is also known as the pluricomplex Green function with logarithmic growth or a logarithmic pole at infinity. We extend Lempert's formula for this function from the convex case to the connected case.

Let ${\left(f_{\lambda }\right)}_{\lambda \in \Lambda }$ be a holomorphic family of rational mappings of degree $d$ on ${\mathbb{P}}^1\left(ℂ\right)$, with $k$ marked critical points $c_1,...,c_k$. To this data is associated a closed positive current $T_1\wedge \cdots \wedge T_k$ of bidegree $(k,k)$ on $\Lambda$, aiming to describe the simultaneous bifurcations of the marked critical points. In this note we show that the support of this current is accumulated by parameters at which $c_1,...,c_k$ eventually fall on repelling cycles. Together with results of Buff, Epstein and Gauthier, this leads to a complete characterization of $\mathrm{Supp}(T_1\wedge \cdots \wedge T_k)$.

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 $T_d:=x\in {ℝ}_{+}^d:{\sum }_{j=1}^d{x}_j\le 1.$

On définit sur un espace vectoriel $E$ une classe de topologies qui rendent la multiplication continue, mais ne sont pas vectorielles en général. Sur un espace complexe $E$ elles permettent d’obtenir encore les principales propriétés des fonctions plurisousharmoniques. De telles topologies séparées sont localement pseudo-convexes (mais non localement convexes en général) : cette notion intervient dans les extensions données récemment par l’auteur du théorème de Banach-Steinhaus aux familles de polynômes...