Phase space transforms and microlocal analysis. (English) Zbl 1111.35143
Colombini, Ferruccio (ed.) et al., Phase space analysis of partial differential equations. Vol. II. Pisa: Scuola Normale Superiore (ISBN 88-7642-150-5/pbk). Pubblicazioni del Centro di Ricerca Matematica Ennio de Giorgi, 505-524 (2004).
The author introduces a phase space approach to microlocal analysis by means of the Bargmann transform allowing the representation of functions as smooth superposition of elementary pieces or coherent states which are strongly localized in position and frequency. A simple characterization of \(S_{00}^0\) type pseudodifferential operators is given. Higher order calculus is introduced to prove the sharp Gårding inequality and the Fefferman-Phong inequality. \(S_{00}^0\) type Fourier integral operators associated with bilipschitz canonical transformations are introduced. It is shown that evolutions operators are Fourier integral operators associated with the Hamilton flow maps. A Egorov theorem is also given.
For the entire collection see [Zbl 1099.35003].
For the entire collection see [Zbl 1099.35003].
Reviewer: Rémi Vaillancourt (Ottawa)
MSC:
| 35S05 | Pseudodifferential operators as generalizations of partial differential operators |
| 35S30 | Fourier integral operators applied to PDEs |
| 26D10 | Inequalities involving derivatives and differential and integral operators |
