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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The pluricomplex Poisson kernel for strongly convex domains


Authors: Filippo Bracci, Giorgio Patrizio and Stefano Trapani
Journal: Trans. Amer. Math. Soc. 361 (2009), 979-1005
MSC (2000): Primary 32W20, 32U35
Published electronically: August 18, 2008
MathSciNet review: 2452831
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Abstract: Let $ D$ be a bounded strongly convex domain in the complex space of dimension $ n$. For a fixed point $ p\in \partial 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.


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Additional Information

Filippo Bracci
Affiliation: Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Roma, Italy.
Email: fbracci@mat.uniroma2.it

Giorgio Patrizio
Affiliation: Dipartimento di Matematica “Ulisse Dini”, Università di Firenze, Viale Morgagni 67-A, 50134 Firenze, Italy.
Email: patrizio@math.unifi.it

Stefano Trapani
Affiliation: Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Roma, Italy.
Email: trapani@mat.uniroma2.it

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04549-2
PII: S 0002-9947(08)04549-2
Received by editor(s): September 26, 2006
Received by editor(s) in revised form: May 2, 2007
Published electronically: August 18, 2008
Article copyright: © Copyright 2008 American Mathematical Society