The pluricomplex Poisson kernel for convex finite‐type domains

Abstract

Given a bounded convex domain of finite D'Angelo type and a boundary point , we prove that the homogeneous complex Monge–Ampère equation  possesses a continuous strictly negative solution that vanishes on and has a simple pole at . We establish that equals (up to sign) the normal derivative at of the pluricomplex Green function , and its sublevel sets are the horospheres centered at . Moreover, is the maximal element of the family of psh functions with a prescribed behavior at , that is, it satisfies a Phragmen–Lindelöf type theorem and provides a reproducing formula for plurisubharmonic functions. Consequently, serves as a generalization of the classical Poisson kernel of the unit disc. Our approach, based on metric methods and scaling techniques, allows our results to be applied to strongly convex domains with ‐smooth boundaries as well. In the course of the proof, we also establish a novel estimate of the Kobayashi distance near boundary points.