Abstract
AbstractIn this paper we use some ideas from [12, 13] and consider the description of Hörmander type pseudo-differential operators on $$\mathbb {R}^d$$
R
d
($$d\ge 1$$
d
≥
1
), including the case of the magnetic pseudo-differential operators introduced in [15, 16], with respect to a tight Gabor frame. We show that all these operators can be identified with some infinitely dimensional matrices whose elements are strongly localized near the diagonal. Using this matrix representation, one can give short and elegant proofs to classical results like the Calderón-Vaillancourt theorem and Beals’ commutator criterion, and also establish local trace-class criteria.
Publisher
Springer Science and Business Media LLC
Reference21 articles.
1. Arsu, G.: On Schatten-von Neumann class properties of pseudo-differential operators. The Cordes-Kato method. J. Operator Theory. 55(1), 81–114 (2008)
2. Beals, R.: Characterization of pseudodifferential operators and applications. Duke Math. J. 44(1), 45–57 (1977)
3. J. M. Bony: Caractérisation des opérateurs pseudo-différentiels. École Polytechnique, Séminaire E.D.P., Exposé no. XXIII (1996–1997)
4. Bony, J.M., Chemin, Y.: Espaces functionels associés au calcul de Weyl-Hörmander. Bull Soc. Math. France 122, 77–118 (1994)
5. Bony, J.M., Lerner, N.: Quantification asymptotique et microlocalisations d’ordre supérieur. I. Ann. Sci. Ec. Norm. Super. (4) 22(3), 377–433 (1989)