The pluricomplex Poisson kernel for convex finite type domains

2025/09/30 by Leandro Arosio, Filippo Bracci, Arosio, Leandro +3
#math.CV

paper · pdf · doi:10.48550/arxiv.2509.26230

Abstract

Given a bounded convex domain D⊂ \mathbb Cⁿ of finite D'Angelo type and a boundary point ξ∈ ∂ D, we prove that the homogeneous complex Monge-Ampère equation (dd^cu)ⁿ=0 possesses a continuous strictly negative solution Ω_ξ that vanishes on ∂ D∖ \ξ\ and has a simple pole at ξ. We establish that Ω_ξ(z) equals (up to sign) the normal derivative at ξ of the pluricomplex Green function G_z, and its sublevel sets are the horospheres centered at ξ. Moreover, Ω_ξ satisfies a Phragmen-Lindelöf type-theorem and provides a reproducing formula for plurisubharmonic functions. Consequently, Ω_ξ 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 C²-smooth boundaries as well. In the course of the proof, we also establish a novel estimate of the Kobayashi distance near boundary points.

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