The pluricomplex Poisson kernel for strongly convex domains

Details The pluricomplex Poisson kernel for strongly convex domains The pluricomplex Poisson kernel for strongly convex domains

2005-07-12

arxiv.org/abs/math/0507247v2

Mathematics

Complex Variables

31 pages

Scientific paper

Let $D$ be a bounded strongly convex domain in the complex space of dimension $n$. Fixed a point $p\in \partial D$, we consider the solution of a homogeneous complex Monge-Ampere equation with 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 and reproducing formulas for plurisubharmonic functions.

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