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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The geometry of analytic varieties satisfying the local Phragmén-Lindelöf condition and a geometric characterization of the partial differential operators that are surjective on $\mathcal{A}(\mathbb{R} ^4)$


Authors: Rüdiger W. Braun, Reinhold Meise and B. A. Taylor
Journal: Trans. Amer. Math. Soc. 356 (2004), 1315-1383
MSC (2000): Primary 32C25; Secondary 32U05, 35E10
DOI: https://doi.org/10.1090/S0002-9947-03-03448-2
Published electronically: October 21, 2003
MathSciNet review: 2034311
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Abstract: The local Phragmén-Lindelöf condition for analytic subvarieties of  $\mathbb{C} ^n$ at real points plays a crucial role in complex analysis and in the theory of constant coefficient partial differential operators, as Hörmander has shown. Here, necessary geometric conditions for this Phragmén-Lindelöf condition are derived. They are shown to be sufficient in the case of curves in arbitrary dimension and of surfaces in  $\mathbb{C} ^3$. The latter result leads to a geometric characterization of those constant coefficient partial differential operators which are surjective on the space of all real analytic functions on  $\mathbb{R} ^4$.


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Additional Information

Rüdiger W. Braun
Affiliation: Mathematisches Institut, Heinrich-Heine-Universität, Universitätsstraße 1, 40225 Düsseldorf, Germany
Email: Ruediger.Braun@uni-duesseldorf.de

Reinhold Meise
Affiliation: Mathematisches Institut, Heinrich-Heine-Universität, Universitätsstraße 1, 40225 Düsseldorf, Germany
Email: meise@cs.uni-duesseldorf.de

B. A. Taylor
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
Email: taylor@umich.edu

DOI: https://doi.org/10.1090/S0002-9947-03-03448-2
Received by editor(s): July 12, 2002
Published electronically: October 21, 2003
Additional Notes: The authors gratefully acknowledge support of DAAD and NSF under the program “Projektbezogene Förderung des Wissenschaftleraustausch mit den USA in Zusammenarbeit mit der National Science Foundation” and of the Volkswagen-Stiftung (RiP-program in Oberwolfach). The research of the third-named author was supported in part by the National Science Foundation under grant number DMS 0070725.
Article copyright: © Copyright 2003 American Mathematical Society