Microlocalization at infinity of the sheaf of real analytic functions (Q1060481)

Let \({\mathbb{D}}^ n\) be the radial compactification of \({\mathbb{R}}^ n\), i.e., \({\mathbb{D}}^ n={\mathbb{R}}^ n\cup S^{n-1}\) where \(S^{n-1}\) is the n-1 dimensional sphere consisting of infinite length vectors in \({\mathbb{R}}^ n\). Let f(x) be a section of real analytic functions on \({\mathbb{R}}^ n\). Then f(x) is naturally regarded as a function in \({\mathbb{D}}^ n\) with singularity at \(S^{n-1}\). The purpose of this paper is to microlocalize such function f(x), particularly, the singularity of f(x) at infinity in the sphere \(S^{n-1}\). The author constructs a sheaf \({\mathcal C}^{\infty}\) on the cosphere bundle \(S^*{\mathbb{D}}^ n\) of \({\mathbb{D}}^ n\) and proves a fundamental exact sequence: \[ (*)\quad 0\to (\iota_*{\mathcal O})|_{{\mathbb{D}}^ n}\to^{b}\iota_*{\mathcal A}\to^{sp}\pi '\!_*{\mathcal C}^{\infty}\to 0. \] Here, \({\mathcal O}\) is the sheaf of holomorphic functions on \({\mathbb{C}}^ n:\) \({\mathcal A}\) is the sheaf of real analytic functions on \({\mathbb{R}}^ n:\) \(\iota\) is the natural injection \({\mathbb{C}}^ n\to {\mathbb{D}}^ n+\sqrt{-1}{\mathbb{R}}^ n:\) \(\pi\) ' the natural projection \(S^*{\mathbb{D}}^ n\to {\mathbb{D}}^ n\). In the sequence (*), \((\iota_*{\mathcal O})\) (resp. \(\iota_*({\mathcal A}),{\mathcal C}^{\infty})\) corresponds to the sheaf \({\mathcal A}\) on \({\mathbb{R}}^ n\) (resp. the sheaf of hyperfunctions \({\mathcal B}\) on \({\mathbb{R}}^ n\), the sheaf of microfunctions \({\mathcal C}\) on \(S^*{\mathbb{R}}^ n)\) in ordinary hyperfunction theory. Moreover, the author defines the wave front set at infinity: \(W.F.^{\infty}(f):=\sup p_{{\mathcal C}^{\infty}}(sp(f))\) for a section of \(\iota_*({\mathcal A})\), and shows several parallel theorems in the hyperfunction theory.