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On bounded pseudodifferential operators in a high-dimensional setting


Authors: L. Amour, L. Jager and J. Nourrigat
Journal: Proc. Amer. Math. Soc. 143 (2015), 2057-2068
MSC (2010): Primary 35S05
DOI: https://doi.org/10.1090/S0002-9939-2014-12379-3
Published electronically: December 22, 2014
MathSciNet review: 3314115
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Abstract | References | Similar Articles | Additional Information

Abstract: This work is concerned with extending the results of Calderón and Vaillancourt, proving the boundedness of Weyl pseudodifferential operators $ Op_h^{Weyl} (F)$ in $ L^2({\mathbb{R}}^n)$. We state conditions under which the norm of such operators has an upper bound independent of $ n$. To this aim, we apply a decomposition of the identity to the symbol $ F$, thus obtaining a sum of operators of a hybrid type, each of them behaving as a Weyl operator with respect to some of the variables and as an anti-Wick operator with respect to the other ones. Then we establish upper bounds for these auxiliary operators, using suitably adapted classical methods like coherent states.


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Additional Information

L. Amour
Affiliation: Laboratoire de Mathématiques, EA 4535, FR CNRS-3399, Université de Reims Champagne-Ardenne, 51687 Reims, France
Email: laurent.amour@univ-reims.fr

L. Jager
Affiliation: Laboratoire de Mathématiques, EA 4535, FR CNRS-3399, Université de Reims Champagne-Ardenne, 51687 Reims, France
Email: lisette.jager@univ-reims.fr

J. Nourrigat
Affiliation: Laboratoire de Mathématiques, EA 4535, FR CNRS-3399, Université de Reims Champagne-Ardenne, 51687 Reims, France
Email: jean.nourrigat@univ-reims.fr

DOI: https://doi.org/10.1090/S0002-9939-2014-12379-3
Received by editor(s): March 11, 2013
Received by editor(s) in revised form: July 13, 2013, September 6, 2013, September 19, 2013, and October 9, 2013
Published electronically: December 22, 2014
Dedicated: Dedicated to the memory of Bernard Lascar
Communicated by: Michael Hitrik
Article copyright: © Copyright 2014 American Mathematical Society