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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
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