The pluricomplex Poisson Kernel for strongly convex domain.

Abstract. Let D be a bounded strongly convex domain in the complex space of dimension n. For a fixed point p ∈ ∂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.