## The FBI transform on compact $${\mathcal{C}^\infty}$$ manifolds.(English)Zbl 0974.35005

This paper can be considered as a comprehensive geometric treatment of Cordoba and Fefferman theory of wave-packet integral kernels; see A. Córdoba and C. Fefferman [Commun. Partial Differ. Equations 3, 979-1005 (1978; Zbl 0389.35046)]. In fact, a geometric theory of the FBI transform and a compact $$C^\infty$$ manifold is presented, as the FBI transform is considered as a generalization of the classical notion of the wave-packet transform.
The authors point of view is similar to those of J. Sjöstrand [Can. J. Math. 48, No. 2, 397-447 (1996; Zbl 0863.35071)], but dropping analyticity assumption and constructing directly the orthogonal projection onto the range of the transform. An interconnection with Melrose’s “scattering calculus” of pseudodifferential operators on the noncompact manifold $$T^*M$$ is described, too.

### MSC:

 35A22 Transform methods (e.g., integral transforms) applied to PDEs 58J40 Pseudodifferential and Fourier integral operators on manifolds 81R30 Coherent states

### Keywords:

wave packet; scattering calculus

### Citations:

Zbl 0389.35046; Zbl 0863.35071
Full Text:

### References:

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