The pluricomplex Poisson kernel for strongly convex domains

Abstract:

Let $D$ be a bounded strongly convex domain in the complex space of dimension $n$. For a fixed point $p\in \partial D$, we consider the solution of a homogeneous complex Monge-Ampère equation with a simple pole at $p$. We prove that such a solution enjoys many properties of the classical Poisson kernel in the unit disc and thus deserves to be called the pluricomplex Poisson kernel of $D$ with pole at $p$. In particular we discuss extremality properties (such as a generalization of the classical Phragmen-Lindelof theorem), relations with the pluricomplex Green function of $D$, uniqueness in terms of the associated foliation and boundary behaviors. Finally, using such a kernel we obtain explicit reproducing formulas for plurisubharmonic functions.