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....