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Théorie de la deuxième microlocalisation dans le domaine complexe. (English) Zbl 0561.32013
Progress in Mathematics, Vol. 53. Boston-Basel-Stuttgart: Birkhäuser. XVI, 311 p. DM 82.00 (1985).
This book is an original research monograph whose aim is to iterate the construction of the microdifferential operators as done by M. Sato, T. Kawai and M. Kashiwara [in Hyperfunctions pseudodiff. Equations, Proc. Conf. Katata 1971, Lect. Notes Math. 287, 263-529 (1973; Zbl 0277.46039)]. Namely, let X be a complex analytic manifold, \(\pi: T^*X\to X\) its cotangent bundle, and \({\mathcal D}_ X\) its sheaf of differential operators of finite order with analytic coefficients. Then Sato, Kawai and Kashiwada constructed the sheaf \({\mathcal E}_ X\) of ”microdifferential operators” on \(T^*X\) such that \(P\in \pi^{-1}{\mathcal D}_ X\) is invertible in \({\mathcal E}_ X\) outside its characteristic variety. Now, let \(\Lambda\) be a homogeneous involutive submanifold of \(T^*X\), \({\tilde \Lambda}\) its union of bicharacteristic leaves, and the bundle \(\pi: T^*_{\Lambda}{\tilde \Lambda}\to \Lambda\). Then the author succeeds to use the sheaves defined by Sato, Kawai and Kashiwada instead of the structural sheaf \({\mathcal O}_ X\) such as to construct a sheaf \({\mathcal E}^ 2_{\Lambda}\) of 2-microdifferential operators on \(T^*_{\Lambda}{\tilde \Lambda}\) with the property that \(P\in \pi^{- 1}({\mathcal E}_ X/\Lambda)\) is invertible in \({\mathcal E}^ 2_{\Lambda}\) outside its microcharacteristic variety. Then he develops a symbol theory (involving double series of holomorphic functions), and studies sheaves \({\mathcal E}^ 2_{\Lambda}(r,s)\) for rational numbers r,s with \(1\leq s\leq r\leq +\infty\) which generalize \({\mathcal E}^ 2_{\Lambda}={\mathcal E}^ 2_{\Lambda}(1,1)\). All these objects are used in the third chapter of the book for applications to systms of differential and microdifferential equations which include a generalization of the Newton polygon, a generalization of the Levi conditions, convergence results for formal series solutions, a study of the Cauchy problem in the complex domain etc.
Reviewer: I.Vaisman

32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces
58-02 Research exposition (monographs, survey articles) pertaining to global analysis
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
58J10 Differential complexes
32Q99 Complex manifolds
47Gxx Integral, integro-differential, and pseudodifferential operators
32K15 Differentiable functions on analytic spaces, differentiable spaces
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials