The pluricomplex Poisson kernel for strongly convex domains
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- by Filippo Bracci, Giorgio Patrizio and Stefano Trapani PDF
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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.References
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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.
- MR Author ID: 631111
- 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
- Received by editor(s): September 26, 2006
- Received by editor(s) in revised form: May 2, 2007
- Published electronically: August 18, 2008
- © Copyright 2008 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 361 (2009), 979-1005
- MSC (2000): Primary 32W20, 32U35
- DOI: https://doi.org/10.1090/S0002-9947-08-04549-2
- MathSciNet review: 2452831