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On the anti-Wick symbol as a Gelfand-Shilov generalized function


Authors: L. Amour, N. Lerner and J. Nourrigat
Journal: Proc. Amer. Math. Soc. 148 (2020), 2909-2914
MSC (2010): Primary 47G30, 46F05
DOI: https://doi.org/10.1090/proc/14933
Published electronically: February 26, 2020
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Abstract: The purpose of this article is to prove that the anti-Wick symbol of an operator mapping $ \mathcal {S}(\mathbb{R}^n)$ into $ \mathcal {S}'(\mathbb{R}^n)$, which is generally not a tempered distribution, can still be defined as a Gelfand-Shilov generalized function. This result relies on test function spaces embeddings involving the Schwartz and Gelfand-Shilov spaces. An additional embedding concerning Schwartz and Gevrey spaces is also given.


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Additional Information

L. Amour
Affiliation: LMR, Université de Reims Champagne-Ardenne, Moulin de la Housse, BP 1039, 51687 REIMS Cedex 2, France
Address at time of publication: LMR FRE CNRS 2011, Université de Reims, France
Email: laurent.amour@univ-reims.fr

N. Lerner
Affiliation: IMJ-PRG, Sorbonne Université, Campus Pierre et Marie Curie, 4 place Jussieu, 75252 Paris Cedex, France
Address at time of publication: IMJ UMR CNRS 7586, Sorbonne Université, France
Email: nicolas.lerner@imj-prg.fr

J. Nourrigat
Affiliation: LMR, Université de Reims Champagne-Ardenne, Moulin de la Housse, BP 1039, 51687 REIMS Cedex 2, France
Address at time of publication: LMR FRE CNRS 2011, Université de Reims, France
Email: jean.nourrigat@univ-reims.fr

DOI: https://doi.org/10.1090/proc/14933
Keywords: Anti-Wick symbol, Gelfand-Shilov generalized functions, pseudodifferential calculus, Gevrey spaces, test function spaces embeddings.
Received by editor(s): May 24, 2019
Received by editor(s) in revised form: November 4, 2019
Published electronically: February 26, 2020
Communicated by: Ariel Barton
Article copyright: © Copyright 2020 American Mathematical Society