Abstract. Let D be a bounded strongly convex domain in the complex space of dimension n. For a fixed point p ∈ ∂D, we consider the solution of a homogeneous complex Monge-Ampère 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.
The pluricomplex Poisson Kernel for strongly convex domain / G. PATRIZIO; F. BRACCI; S. TRAPANI. - In: TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY. - ISSN 0002-9947. - STAMPA. - 361:(2009), pp. 979-1005. [10.1090/S0002-9947-08-04549-2]
The pluricomplex Poisson Kernel for strongly convex domain.
PATRIZIO, GIORGIO;
2009
Abstract
Abstract. Let D be a bounded strongly convex domain in the complex space of dimension n. For a fixed point p ∈ ∂D, we consider the solution of a homogeneous complex Monge-Ampère 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.File | Dimensione | Formato | |
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