The pluricomplex Poisson kernel for convex finite type domains

Abstract

Given a bounded convex domain $D\subset \mathbb C^n$ of finite D'Angelo type and a boundary point $\xi\in \partial D$, we prove that the homogeneous complex Monge-Amp\`ere equation $(dd^cu)^n=0$ possesses a continuous strictly negative solution $\Omega_\xi$ that vanishes on $\partial D\setminus \{\xi\}$ and has a simple pole at $\xi$. We establish that $\Omega_\xi(z)$ equals (up to sign) the normal derivative at $\xi$ of the pluricomplex Green function $G_z$, and its sublevel sets are the horospheres centered at $\xi$. Moreover, $\Omega_\xi$ satisfies a Phragmen-Lindel\"of type-theorem and provides a reproducing formula for plurisubharmonic functions. Consequently, $\Omega_\xi$ serves as a generalisation 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 $C^2$-smooth boundaries as well. In the course of the proof, we also establish a novel estimate of the Kobayashi distance near boundary points.