We give some characterizations of the class ${}_m\left(\Omega \right)$ and use them to establish a lower estimate for the log canonical threshold of plurisubharmonic functions in this class.

We find a bounded solution of the non-homogeneous Monge-Ampère equation under very weak assumptions on its right hand side.

It is described how both plurisubharmonicity and convexity of functions can be characterized in terms of simple to work with classes of holomorphic martingales, namely a class of driftless Itô processes satisfying a skew-symmetry property and a family of linear modifications of Brownian motion parametrized by a compact set.

Let $A_{\zeta }=\Omega -\overline{\rho \left(\zeta \right)·\Omega }$ be a family of generalized annuli over a domain U. We show that the logarithm of the Bergman kernel $K_{\zeta }\left(z\right)$ of $A_{\zeta }$ is plurisubharmonic provided ρ ∈ PSH(U). It is remarkable that $A_{\zeta }$ is non-pseudoconvex when the dimension of $A_{\zeta }$ is larger than one. For standard annuli in ℂ, we obtain an interesting formula for $\partial ²log{K}_{\zeta }/\partial \zeta \partial \zeta ̅$, as well as its boundary behavior.

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.

Let $D$ be a pseudoconvex domain in ${ℂ}_t^k\times {ℂ}_z^n$ and let $\phi$ be a plurisubharmonic function in $D$. For each $t$ we consider the $n$-dimensional slice of $D$, $D_t=\{z;(t,z)\in D\}$, let ${\phi }^t$ be the restriction of $\phi$ to $D_t$ and denote by $K_t(z,\zeta )$ the Bergman kernel of $D_t$ with the weight function ${\phi }^t$. Generalizing a recent result of Maitani and Yamaguchi (corresponding to $n=1$ and $\phi =0$) we prove that $log{K}_t(z,z)$ is a plurisubharmonic function in $D$. We also generalize an earlier results of Yamaguchi concerning the Robin function and discuss similar results in the setting...

Let $\mathrm{\Sigma }$ be the boundary of the unit ball $\mathrm{\Omega }$ of ${𝐂}^n$. A set of second order linear partial differential operators, tangential to $\mathrm{\Sigma }$, is explicitly given in such a way that, for $n\ge 3$, the corresponding PDE caractherize the trace of the solution of the pluriharmonic problem (either “in the large” or “local”), relative to $\mathrm{\Omega }$.

0. Introduction. Nous donnons ici une étude systématique des systèmes doublement orthogonaux "de Bergman" et leurs applications à certains aspects de l'analyse pluricomplexe: espaces de fonctions holomorphes, fonctions séparément analytiques. C'est en quelque sorte un article de synthèse. On y trouve cependant des démonstrations détaillées qui n'ont paru nulle part ailleurs.

We establish the comparison principle in the class ${}_p\left(f\right)$. The result obtained is applied to the Dirichlet problem in ${}_p\left(f\right)$.

Let Ω be a bounded convex domain in Rn with smooth, strictly convex boundary ∂Ω, i.e. the principal curvatures of ∂Ω are all positive. We study the problem of finding a convex function u in Ω such that:det (uij) = 0 in Ωu = φ given on ∂Ω.