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Transactions of the American Mathematical Society

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From semigroups to subelliptic estimates for quadratic operators


Authors: Michael Hitrik, Karel Pravda-Starov and Joe Viola
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 35A22, 35H20, 47D06, 53D22
DOI: https://doi.org/10.1090/tran/7251
Published electronically: May 17, 2018
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Abstract: Using an approach based on the techniques of FBI transforms, we give a new simple proof of the global subelliptic estimates for non-self-adjoint nonelliptic quadratic differential operators, under a natural averaging condition on the Weyl symbols of the operators, established by the second author in Subelliptic estimates for quadratic differential operators (Amer. J. Math. 133 (2011), no. 1, 39-89). The loss of the derivatives in the subelliptic estimates depends directly on algebraic properties of the Hamilton maps of the quadratic symbols. Using the FBI point of view, we also give accurate smoothing estimates of Gelfand-Shilov type for the associated heat semigroup in the limit of small times.


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Additional Information

Michael Hitrik
Affiliation: Department of Mathematics, University of California, Los Angeles, California
Email: hitrik@math.ucla.edu

Karel Pravda-Starov
Affiliation: IRMAR, CNRS UMR 6625, Université de Rennes 1, Campus de Beaulieu, 263 avenue du Général Leclerc, CS 74205, 35042 Rennes Cedex, France
Email: karel.pravda-starov@univ-rennes1.fr

Joe Viola
Affiliation: Laboratoire de Mathématiques Jean Leray, Université de Nantes, rue de la Houssinière, BP 92208, 44322 Nantes Cedex 3, France
Email: joseph.viola@univ-nantes.fr

DOI: https://doi.org/10.1090/tran/7251
Keywords: Non-self-adjoint operator, subelliptic estimate, quadratic differential operator, heat semigroup, FBI transform
Received by editor(s): January 25, 2016
Received by editor(s) in revised form: September 5, 2016, and March 28, 2017
Published electronically: May 17, 2018
Additional Notes: The research of the second and third authors is supported by the ANR NOSEVOL (Project: ANR 2011 BS0101901).
Article copyright: © Copyright 2018 American Mathematical Society

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