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Extremal bases, geometrically separated domains and applications

Authors: Ph. Charpentier and Y. Dupain
Original publication: Algebra i Analiz, tom 26 (2014), nomer 1.
Journal: St. Petersburg Math. J. 26 (2015), 139-191
MSC (2010): Primary 32T25; Secondary 32T27
Published electronically: November 21, 2014
MathSciNet review: 3234809
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Abstract: The notion of an extremal basis of tangent vector fields is introduced for a boundary point of finite type of a pseudo-convex domain in $ \mathbb{C}^n$, $ n\geq 3$. By using this notion, the class of geometrically separated domains at a boundary point is defined and a description of their complex geometry is presented. Examples of such domains are given, for instance, by locally lineally convex domains, domains with locally diagonalizable Levi form at a point, or by domains for which the Levi form has comparable eigenvalues near a point. Moreover, it is shown that geometrically separated domains can be localized. An example of a not geometrically separated domain is presented. Next, the so-called ``adapted plurisubharmonic functions'' are defined and sufficient conditions, related to extremal bases, for their existence are given. Then, for these domains, when such functions exist, global and local sharp estimates are proved for the Bergman and Szegő projections. As an application, a result by C. Fefferman, J. J. Kohn, and M. Machedon for the local Hölder estimate of the Szegő projection is refined, by removing the arbitrarily small loss in the Hölder index and giving a stronger nonisotropic estimate.

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Additional Information

Ph. Charpentier
Affiliation: Université Bordeaux 1, Institut de Mathématiques, 351 cours de la Libération, 33405 Talence, France

Y. Dupain
Affiliation: Université Bordeaux 1, Institut de Mathématiques, 351 cours de la Libération, 33405 Talence, France

Keywords: Finite type, extremal basis, complex geometry, adapted plurisubharmonic function, Bergman and Szeg\H{o} projections
Received by editor(s): September 24, 2012
Published electronically: November 21, 2014
Article copyright: © Copyright 2014 American Mathematical Society