Regularity theorems for holonomic modules (Q1320647)

Let \(X\) be a complex manifold, \(T^* X\) its cotangent bundle, \(\overset\circ T^*= T^*\backslash T_ X X\), \({\mathcal E}^ \infty_ X\) the sheaf of micro-differential operators of infinite order and \({\mathcal E}_ X\) (resp. \({\mathcal E}_ X(m)\)) the subsheaf of \({\mathcal E}^ \infty_ X\) consisting of micro-differential operators of finite order (resp. order \(\leq m\)). The sheaf \({\mathcal E}^{(s)}_ X\) of micro- differential operators of Gevrey growth of order \((s)\) is introduced \((s\in (1,\infty))\). Next for \(\sigma\in [1,\infty)\) a rational number of \(\mathcal M\) a holonomic \({\mathcal E}_ X\) module the weak irregularity at most \(\sigma\) of \(\mathcal M\) is defined. The main result is: Theorem. Let \(U\subset T^* X\) be a \(\mathbb{C}^*\) conic open set, \({\mathcal M}\) a holonomic \({\mathcal E}_ X\) module on \(U\) and \(\sigma\geq 1\) a rational number. The following are equivalent: (i) There exists a holonomic \({\mathcal E}_ X\) module \({\mathcal M}_{\text{reg}}\) with regular singularities satisfying: \({\mathcal E}^{(s)}_ X\otimes_{{\mathcal E}_ X}{\mathcal M}= {\mathcal E}^{(s)}_ X\otimes {\mathcal M}_{\text{reg}}\) in \(U\) for all \(s\in [1,\sigma/(\sigma- 1)]\), (ii) for any submanifold \(Y\subset X\) and \(s\in [1,\sigma/(\sigma-1)]\) we have: \({\mathcal R}\Hom_{{\mathcal E}_ X}({\mathcal M},{\mathcal C}^{{\mathcal R},(s)}_{Y| X})|_ U= {\mathcal R}\Hom_{{\mathcal E}_ X}({\mathcal M},{\mathcal C}^{{\mathcal R}}_{Y| X})|_ U\), (iii) \(\mathcal M\) has weak irregularity at most \(\sigma\) in \(U\). Above \({\mathcal C}^{{\mathcal R},(s)}_{Y| X}\) is the subsheaf of the holomorphic microfunctions \({\mathcal C}^{{\mathcal R}}_{Y| X}\) defined by \({\mathcal C}^{{\mathcal R},(s)}_{Y| X}= {\mathcal E}^{(s)}_ X {\mathcal C}^{{\mathcal R},f}_{Y| X}\), where \({\mathcal C}^{{\mathcal R},f}_{Y| X}\) is the subsheaf of tempered holomorphic microfunctions.