Abstract
In the framework of the Weyl–Hörmander calculus, under a condition of “geodesic temperance”, pseudodifferential operators can be characterized by the boundedness of their iterated commutators. As a corollary, functions of pseudodifferential operators are themselves pseudodifferential. Sufficient conditions are given for the geodesic temperance. In particular, it is valid in the Beals–Fefferman calculus.
2010 Mathematics Subject Classification:35S05, 47A60, 47G30.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Beals, R.: Characterization of pseudodifferential operators and applications. Duke Math. J. 44(1), 45–57 (1977)
Beals, R., Fefferman, C.: Spatially inhomogeneous pseudodifferential operators I. Comm. Pure Appl. Math. 27, 1–24 (1974)
Bony, J.-M.: Caractérisations des opérateurs pseudo-différentiels. In: Séminaire Équations aux Dérivées Partielles 96–97, Les Éditions de l’École Polytechnique, F-91128 Palaiseau cedex, France, Exp. no. XXIII, pp. 17 (1996–1997) ISBN: 2-7302-0396-6
Bony, J.-M.: Evolution equations and microlocal analysis. In: Hyperbolic Problems and Related Topics, pp. 17–40. Grad. Ser. Anal. International Press, Somerville (2003)
Bony, J.-M., Chemin, J.-Y.: Espaces fonctionnels associés au calcul de Weyl–Hörmander. Bull. Soc. Math. France 122(1), 77–118 (1994)
Hörmander, L.: The Analysis of Linear Partial Differential Operators. Springer, Berlin (1990)
Toft, J.: Continuity and Schatten properties for pseudo-differential operators on modulation spaces. In: Modern Trends in Pseudo-Differential Operators. Operational Theory in Advanced Application, vol. 172, pp. 173–206. Birkhäuser, Basel (2007)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media New York
About this chapter
Cite this chapter
Bony, JM. (2013). On the Characterization of Pseudodifferential Operators (Old and New). In: Cicognani, M., Colombini, F., Del Santo, D. (eds) Studies in Phase Space Analysis with Applications to PDEs. Progress in Nonlinear Differential Equations and Their Applications, vol 84. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4614-6348-1_2
Download citation
DOI: https://doi.org/10.1007/978-1-4614-6348-1_2
Published:
Publisher Name: Birkhäuser, New York, NY
Print ISBN: 978-1-4614-6347-4
Online ISBN: 978-1-4614-6348-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)