Multi-specialization and multi-asymptotic expansions

Abstract

In this paper we extend the notion of specialization functor to the case of several closed submanifolds satisfying some suitable conditions. Applying this functor to the sheaf of Whitney holomorphic functions we construct different kinds of sheaves of multi-asymptotically developable functions, whose definitions are natural extensions of the definition of strongly asymptotically developable functions introduced by Majima. Contents1 Multi-normal deformation 5 2 Multi-actions 13 3 Multi-cones 15 4 Multi-normal cones 16 5 Morphisms between multi-normal deformations 24 6 Multi-specialization 33 7 Multi-asymptotic expansions 40 8 Multi-specialization and asymptotic expansions 59 A Conic sheaves 67 B Multi-conic sheaves 69 C Multi-actions in R n 72 D Proof of Proposition 4.7 76References 89Namely, we need the notion of multi-normal deformation and the one of multi-specialization introduced in this paper, which are our main subjects. Let us briefly explain these new notions. Let X be a n-dimensional real analytic manifold with dim X = n, and let χ = {M 1 , . . . , M ℓ } be a family of connected closed submanifolds satisfying some suitable conditions. The multi-normal deformation of X with respect to χ is constructed as follows. We first construct the normal deformation X M 1 of X along M 1 defined by M. Kashiwara and P. Schapira in [10]. Then, taking the pull-back of M 2 in X M 1 , we can obtain the normal deformation X M 1 ,M 2 of X M 1 along the pull-back of M 2 . Then we can define recursively the normal deformation along χ as X = X M 1 ,...,M ℓ :. This manifold is of dimension n + ℓ, it is locally isomorphic to X × R ℓ and in the zero section X × {0} it is isomorphic to × X,1≤j≤ℓ

Multi-microlocalization and microsupport

“…In this section we recall some results of [4]. We first fix some notations, then we recall the notion of multi-normal deformation and the definition of the functor of multi-specialization with some basic properties.…”

“…In [4] the notion of multi-normal deformation was introduced. Here we consider a slight generalization where we replace the condition H2 with the weaker one.…”

“…On the other hand, in the paper [4], the first and the second authors of this article established the notion of the multi-normal cone for a family χ of closed submanifolds with a suitable configuration, and they also constructed the multi-specialization functor along χ. We can observe that cones appearing in simultaneous microlocalization and bi-microlocalization are charac-terized by using the multi-normal cone and that the both microlocalization functors coincide with the multi-microlocalization functor along χ where the latter functor is obtained by repeated application of Sato's Fourier transformation to the multi-specialization functor.…”

“…This shows, in particular, soundness of our framework in the sense that the sharp estimate can be achieved by a geometrical tool (the multi-normal cone) already prepared in our framework. These two results have many applications, and some of them will be given in the last two sections of this paper.The paper is organized as follows: We briefly recall, in Section 1, the theory of the multi-specialization developed in [4]. Then, in Section 2, we define the multi-microlocalization functor by repeatedly applying Sato's Fourier transformation to the multi-specialization functor.…”

“…The paper is organized as follows: We briefly recall, in Section 1, the theory of the multi-specialization developed in [4]. Then, in Section 2, we define the multi-microlocalization functor by repeatedly applying Sato's Fourier transformation to the multi-specialization functor.…”