The pluricomplex Poisson kernel for strongly convex domains

Abstract

Let D be a bounded strongly convex domain in the complex space of dimension n. For a fixed point p epsilon partial derivative D, we consider the solution of a homogeneous complex Monge-Ampere 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

A rigidity condition for generators in strongly convex domains

“…As we remarked in Section 1 (see also [9], [10]), horospheres in a bounded smooth strongly convex domain D are bounded smooth convex subdomain of D; so each point in the boundary of a horosphere has a unique real tangent hyperplane and also a unique complex tangent hyperplane. Hence the complex hyperplane {w ∈ C n | ρ ϕ (w) = z} is the complex tangent hyperplane at ∂E(τ, p, R) in z and thus we can conclude that…”

“…In the unit ball of C n they are ellipsoids internally tangents to the boundary of the unit ball in the center of the horosphere. Recently it has been showed ( [26], [10,Section 4] and [9,Remark 6.4]) that horospheres in bounded smooth strongly convex domains are smooth and strongly convex (and thus strongly pseudoconvex), except at the center and have a global defining function ( [9, Theorem 6.3]). Precisely they are sublevel sets of the pluricomplex Poisson kernel in D (see [10]).…”

Pluripotential theory, semigroups and boundary behavior of infinitesimal generators in strongly convex domains

“…In the papers [12] and [13], the authors prove that u D,p shares many properties with the classical Poisson kernel for the unit disk. In case D = D the unit disc in C, the function…”

“…More information about the properties of u D,p (such as smooth dependence on p, extremality, uniqueness, relations with the pluricomplex Green function, usage in representation formulas for pluriharmonic functions) can be found in [13].…”

A Note on Random Holomorphic Iteration in Convex Domains

“…However, if D is strongly convex then the Lempert projection (that is the one with affine fibers) is unique (see [4,Proposition 3.3]). …”