Higher-codimensional boundary value problems and \(F\)-mild microfunctions -- local and microlocal uniqueness -- (Q1974499)

Hyperfunctions and microfunctions with real analytic parameters are generalized as \(F\)-mild hyperfunctions and \(F\)-mild microfunctions, respectively. More precisely, let \(M\) be a real analytic manifold and \(N\) a closed real analytic submanifold of \(M\) of codimention \(d\geq 2\). By \(X\), and respectively by \(Y\), are denoted the complexification of \(M\), and respectively of \(N\). The purpose is related with the study of higher-codimensional boundary problems for general \(D_X\)-modules, i.e. for systems of linear PDE with analytic coefficients. The main result concerns the local and microlocal uniqueness of \(F\)-mild hyperfunctions or microfunctions solutions of a linear PDE which is Fuchsian along \(Y\) according to I. Lauren and T. Monteiro-Fernandez. Assuming that we have a Fuchs-Goursat system in the sense of the second author, the authors prove the local and microlocal uniqueness of the \(F\)-mild solution for \(D\)-module whose Goursat data are zero. The proofs are based on a sophisticated sheaf's technique elaborated by the authors: several sheaves attached to the boundary are introduced. They are higher-codimensional analogues of those ones defined by the first author [J. Fac. Sci., Univ. Tokyo, Sect. I A 33, 175-232 (1986; Zbl 0626.35058)]. The sheaf of the \(F\)-mild hyperfunction is introduced just here and its microlocalization is developed.