Microlocalization of subanalytic sheaves.

*(English)*Zbl 1295.32018Sheaves on the subanalytic site are included in the category of ind-sheaves, introduced by M. Kashiwara and P. Schapira, but there are advantages in to study this subcategory by itself, since it is more treatable and much more similar to the classic category of sheaves. However, despite the apparent similarity with classical sheaves, the techniques used in the framework of subanalytic sheaves are (even) much more finest and sophisticated.

The author has studied exhaustively the category of subanalytic sheaves, by generalizing the most important classical constructions for sheaves. In this paper, he presents a definition for the specialization and microlocalization functors for subanalytic sheaves, which requires the introduction of a suitable category of conic subanalytic sheaves, endowed with a \(\mathbb{R}^+\)-action, as well as the extension of the Fourier-Sato transform.

Interesting applications are exhibited: the functor of specialization appears to be the “key tool in order to give a functorial construction of asymptotically developable functions”, while the functor of microlocalization extend and unifies Andronikof’s and Colin’s well-known “ad hoc” constructions. In particular, the microlocalization of tempered and Whitney holomorphic functions (the prime examples of subanalytic sheaves) is studied.

We also notice that, similarly to all the author’s works, this one is also very detailed and self contained, including an appendix where the relation between ind-sheaves and subanalytic sheaves is recalled and the inverse image of the subanalytic sheaves of tempered and Whitney holomorphic functions is studied.

The author has studied exhaustively the category of subanalytic sheaves, by generalizing the most important classical constructions for sheaves. In this paper, he presents a definition for the specialization and microlocalization functors for subanalytic sheaves, which requires the introduction of a suitable category of conic subanalytic sheaves, endowed with a \(\mathbb{R}^+\)-action, as well as the extension of the Fourier-Sato transform.

Interesting applications are exhibited: the functor of specialization appears to be the “key tool in order to give a functorial construction of asymptotically developable functions”, while the functor of microlocalization extend and unifies Andronikof’s and Colin’s well-known “ad hoc” constructions. In particular, the microlocalization of tempered and Whitney holomorphic functions (the prime examples of subanalytic sheaves) is studied.

We also notice that, similarly to all the author’s works, this one is also very detailed and self contained, including an appendix where the relation between ind-sheaves and subanalytic sheaves is recalled and the inverse image of the subanalytic sheaves of tempered and Whitney holomorphic functions is studied.

Reviewer: Ana Rita Martins (Lisbon)

##### MSC:

32C38 | Sheaves of differential operators and their modules, \(D\)-modules |

35A27 | Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs |

18F20 | Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) |

32B20 | Semi-analytic sets, subanalytic sets, and generalizations |