We generalize classical results for plurisubharmonic functions and hyperconvex domain to q-plurisubharmonic functions and q-hyperconvex domains. We show, among other things, that Bq-regular domains are q-hyperconvex. Moreover, some smoothing results for q-plurisubharmonic functions are also given.

In this note we consider radially symmetric plurisubharmonic functions and the complex Monge-Ampère operator. We prove among other things a complete characterization of unitary invariant measures for which there exists a solution of the complex Monge-Ampère equation in the set of radially symmetric plurisubharmonic functions. Furthermore, we prove in contrast to the general case that the complex Monge-Ampère operator is continuous on the set of radially symmetric plurisubharmonic functions. Finally...

The necessary and sufficient condition that a given plurisubharmonic or a subharmonic function admits the representation by the logarithmic potential (up to pluriharmonic or a harmonic term) is obtained in terms of the Radon transform. This representation is applied to the problem of representation of analytic functions by products of primary factors.

Given a compact set $K\subset {ℂ}^N$, for each positive integer n, let $V^{\left(n\right)}\left(z\right)$ = $V_K^{\left(n\right)}\left(z\right)$ := sup$1/\left(degp\right)V_{p\left(K\right)}\left(p\left(z\right)\right)$: p holomorphic polynomial, 1 ≤ deg p ≤ n. These “extremal-like” functions $V_K^{\left(n\right)}$ are essentially one-variable in nature and always increase to the “true” several-variable (Siciak) extremal function, $V_K\left(z\right)$:= max[0, sup1/(deg p) log|p(z)|: p holomorphic polynomial, ${\left|\right|p\left|\right|}_K\le 1$]. Our main result is that if K is regular, then all of the functions $V_K^{\left(n\right)}$ are continuous; and their associated Robin functions ${\varrho }_{V_K^{\left(n\right)}}\left(z\right):=limsu{p}_{\left|\lambda \right|\to \infty }[V_K^{\left(n\right)}\left(\lambda z\right)-log\left(\right|\lambda \left|\right)]$ increase to ${\varrho }_K:={\varrho }_{V_K}$ for all z outside a pluripolar set....

Let X×Y be the Cartesian product of two locally finite, connected networks that need not have reversible conductance. If X,Y represent random walks, it is known that if X×Y is recurrent, then X,Y are both recurrent. This fact is proved here by non-probabilistic methods, by using the properties of separately superharmonic functions. For this class of functions on the product network X×Y, the Dirichlet solution, balayage, minimum principle etc. are obtained. A unique integral representation is given...

Let E be a compact set in the complex plane, $g_E$ be the Green function of the unbounded component of ${ℂ}_{\infty }\setminus E$ with pole at infinity and $Mₙ\left(E\right)=sup\left(\right||P^{\text{'}}{\left|\right|}_E{)/(\left|\right|P\left|\right|}_E)$ where the supremum is taken over all polynomials ${P|}_E\not\equiv 0$ of degree at most n, and ${\left|\right|f\left|\right|}_E=sup\left|f\right(z\left)\right|:z\in E$. The paper deals with recent results concerning a connection between the smoothness of $g_E$ (existence, continuity, Hölder or Lipschitz continuity) and the growth of the sequence ${Mₙ\left(E\right)}_{n=1,2,...}$. Some additional conditions are given for special classes of sets.

In this paper we investigate some applications of the trace condition for pluriharmonic functions on a smooth, bounded domain in Cn. This condition, related to the normal component on ∂D of the ∂-operator, permits us to study the Neumann problem for pluriharmonic functions and the ∂-problem for (0,1)-forms on D with solutions having assigned real part on the boundary.