Microfunctions at the Boundary and Mild Microfunctions

Abstract

Let $X$ be a real manifold, $\mathcal{F}$ an object of $D^b(X)$, the derived category of the category of bounded complexes of sheaves of abelian groups on $X$. The functor $\mu \mathrm{hom}(\cdot ,\cdot )$, defined in [3], appears to be a useful tool especially in the theory of boundary value problems for partial differential equations. The aim of the present paper is to calculate the stalk of $R{\Gamma }_z\mu \mathrm{hom}({\mathbf{N}}_{\Omega },\mathcal{F})$, when $\Omega$ is a convex (up to diffeomorphism) and open subset of a closed submanifold $M$ of $X$, and $Z$ is a closed convex proper cone of $T^{\ast }X$. As an application we show how to recover in a short and functorial way, the theory of mild microfunctions by Kataoka [5].