The pluricomplex Poisson kernel for strongly convex domains

Abstract

Let D D be a bounded strongly convex domain in the complex space of dimension n n . For a fixed point p ∈ ∂ D p\in \partial D , we consider the solution of a homogeneous complex Monge-Ampère equation with a simple pole at p 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 D with pole at p 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 D , uniqueness in terms of the associated foliation and boundary behaviors. Finally, using such a kernel we obtain explicit reproducing formulas for plurisubharmonic functions.