The functor $beta_{Y}(cdot)$ and mixed problems for $mathcal{D}_{X}$-modules (Microlocal Analysis and Singular Perturbation Theory)

Abstract

Let M be a real analytic manifold, and N be its real analytic submanifold with codimension 1. We denote by X, Y their complexifications. First, we give an algebraic formulation of mixed initial boundary value problems for coherent left DX-modules by using sheaf βY (OX) introduced in [1]. This formulation is of coordinate free because βY (OX) is defined only on X and Y. At the same time, we give a functorial construction of βY (OX). The main results under this formulation are coordinate-free generalizations of our previous results [2] for single differential equations; for example, the estimate of the micro-support of some important solution complex. We will give in [3] the detailed proofs and applications; the existence results of hyperfunction solutions, and the propagation results of micro-analyticity of the solutions along the boundary N as obtained by J. Sjöstrand [4].